Leo-III---1.8.0.UNS-Ref.s

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%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : SWV706-1 : TPTP v9.3.1. Released v4.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39

% Computer : n007.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Sun Sep 27 09:00:15 AM UTC 2026

% Result   : Unsatisfiable 82.58s 34.72s
% Output   : Refutation 90.29s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    2
%            Number of leaves      : 1820
% Syntax   : Number of formulae    : 3641 ( 619 unt;   0 typ;   0 def)
%            Number of atoms       : 10375 (2202 equ;   0 cnn)
%            Maximal formula atoms :    7 (   2 avg)
%            Number of connectives : 70416 (6078   ~;6734   |;   0   &;57604   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   34 (  11 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  110 ( 108 usr;  11 con; 0-11 aty)
%            Number of variables   : 11566 (   0   ^;11566   !;   0   ?;11566   :)

% Comments : 
%------------------------------------------------------------------------------
thf(c_HOL_Oord__class_Oless_decl,type,
    c_HOL_Oord__class_Oless: $i > $i > $i > $o ).

thf(v_i_____decl,type,
    v_i____: $i ).

thf(hAPP_decl,type,
    hAPP: $i > $i > $i ).

thf(c_HOL_Otimes__class_Otimes_decl,type,
    c_HOL_Otimes__class_Otimes: $i > $i ).

thf(tc_nat_decl,type,
    tc_nat: $i ).

thf(c_Int_Onumber__class_Onumber__of_decl,type,
    c_Int_Onumber__class_Onumber__of: $i > $i > $i ).

thf(c_Int_OBit0_decl,type,
    c_Int_OBit0: $i > $i ).

thf(c_Int_OBit1_decl,type,
    c_Int_OBit1: $i > $i ).

thf(c_Int_OPls_decl,type,
    c_Int_OPls: $i ).

thf(c_Power_Opower__class_Opower_decl,type,
    c_Power_Opower__class_Opower: $i > $i ).

thf(v_ka_____decl,type,
    v_ka____: $i ).

thf(c_HOL_Ominus__class_Ominus_decl,type,
    c_HOL_Ominus__class_Ominus: $i > $i > $i > $i ).

thf(c_FFT__Mirabelle_ODFT_decl,type,
    c_FFT__Mirabelle_ODFT: $i > $i > $i > $i ).

thf(c_COMBB_decl,type,
    c_COMBB: $i > $i > $i > $i > $i > $i ).

thf(v_a_____decl,type,
    v_a____: $i ).

thf(tc_Complex_Ocomplex_decl,type,
    tc_Complex_Ocomplex: $i ).

thf(c_FFT__Mirabelle_Oroot_decl,type,
    c_FFT__Mirabelle_Oroot: $i > $i ).

thf(c_Suc_decl,type,
    c_Suc: $i ).

thf(c_FFT__Mirabelle_OFFT_decl,type,
    c_FFT__Mirabelle_OFFT: $i > $i > $i ).

thf(class_OrderedGroup_Oordered__ab__group__add_decl,type,
    class_OrderedGroup_Oordered__ab__group__add: $i > $o ).

thf(tc_Int_Oint_decl,type,
    tc_Int_Oint: $i ).

thf(class_OrderedGroup_Opordered__ab__group__add__abs_decl,type,
    class_OrderedGroup_Opordered__ab__group__add__abs: $i > $o ).

thf(c_lessequals_decl,type,
    c_lessequals: $i > $i > $i > $o ).

thf(c_HOL_Oabs__class_Oabs_decl,type,
    c_HOL_Oabs__class_Oabs: $i > $i > $i ).

thf(c_HOL_Ouminus__class_Ouminus_decl,type,
    c_HOL_Ouminus__class_Ouminus: $i > $i > $i ).

thf(c_HOL_Ozero__class_Ozero_decl,type,
    c_HOL_Ozero__class_Ozero: $i > $i ).

thf(class_Ring__and__Field_Oabs__if_decl,type,
    class_Ring__and__Field_Oabs__if: $i > $o ).

thf(class_Ring__and__Field_Opordered__ring__abs_decl,type,
    class_Ring__and__Field_Opordered__ring__abs: $i > $o ).

thf(class_OrderedGroup_Olordered__ab__group__add_decl,type,
    class_OrderedGroup_Olordered__ab__group__add: $i > $o ).

thf(class_OrderedGroup_Olordered__ab__group__add__abs_decl,type,
    class_OrderedGroup_Olordered__ab__group__add__abs: $i > $o ).

thf(hBOOL_decl,type,
    hBOOL: $i > $o ).

thf(c_Divides_Odiv__class_Odiv_decl,type,
    c_Divides_Odiv__class_Odiv: $i > $i > $i > $i ).

thf(class_Ring__and__Field_Oordered__field_decl,type,
    class_Ring__and__Field_Oordered__field: $i > $o ).

thf(class_Ring__and__Field_Odivision__by__zero_decl,type,
    class_Ring__and__Field_Odivision__by__zero: $i > $o ).

thf(class_Int_Onumber_decl,type,
    class_Int_Onumber: $i > $o ).

thf(c_HOL_Oinverse__class_Odivide_decl,type,
    c_HOL_Oinverse__class_Odivide: $i > $i > $i > $i ).

thf(class_Ring__and__Field_Ofield_decl,type,
    class_Ring__and__Field_Ofield: $i > $o ).

thf(class_Orderings_Opreorder_decl,type,
    class_Orderings_Opreorder: $i > $o ).

thf(c_HOL_Oplus__class_Oplus_decl,type,
    c_HOL_Oplus__class_Oplus: $i > $i ).

thf(c_HOL_Oinverse__class_Oinverse_decl,type,
    c_HOL_Oinverse__class_Oinverse: $i > $i > $i ).

thf(class_Ring__and__Field_Opordered__cancel__semiring_decl,type,
    class_Ring__and__Field_Opordered__cancel__semiring: $i > $o ).

thf(class_Divides_Osemiring__div_decl,type,
    class_Divides_Osemiring__div: $i > $o ).

thf(class_OrderedGroup_Ogroup__add_decl,type,
    class_OrderedGroup_Ogroup__add: $i > $o ).

thf(class_Ring__and__Field_Ocomm__semiring__1_decl,type,
    class_Ring__and__Field_Ocomm__semiring__1: $i > $o ).

thf(c_Ring__and__Field_Oordered__idom_decl,type,
    c_Ring__and__Field_Oordered__idom: $i > $i > $i > $i > $i > $i > $i > $i > $i > $i > $i > $o ).

thf(c_HOL_Oone__class_Oone_decl,type,
    c_HOL_Oone__class_Oone: $i > $i ).

thf(class_OrderedGroup_Opordered__comm__monoid__add_decl,type,
    class_OrderedGroup_Opordered__comm__monoid__add: $i > $o ).

thf(c_Power_Opower_Opower_decl,type,
    c_Power_Opower_Opower: $i > $i > $i > $i ).

thf(c_OrderedGroup_Omonoid__mult_decl,type,
    c_OrderedGroup_Omonoid__mult: $i > $i > $i > $o ).

thf(class_OrderedGroup_Ocomm__monoid__add_decl,type,
    class_OrderedGroup_Ocomm__monoid__add: $i > $o ).

thf(class_OrderedGroup_Opordered__ab__semigroup__add__imp__le_decl,type,
    class_OrderedGroup_Opordered__ab__semigroup__add__imp__le: $i > $o ).

thf(class_Int_Onumber__ring_decl,type,
    class_Int_Onumber__ring: $i > $o ).

thf(class_Orderings_Olinorder_decl,type,
    class_Orderings_Olinorder: $i > $o ).

thf(class_Ring__and__Field_Oidom_decl,type,
    class_Ring__and__Field_Oidom: $i > $o ).

thf(class_RealVector_Oreal__normed__field_decl,type,
    class_RealVector_Oreal__normed__field: $i > $o ).

thf(class_Orderings_Oorder_decl,type,
    class_Orderings_Oorder: $i > $o ).

thf(class_OrderedGroup_Oab__semigroup__mult_decl,type,
    class_OrderedGroup_Oab__semigroup__mult: $i > $o ).

thf(class_OrderedGroup_Opordered__ab__group__add_decl,type,
    class_OrderedGroup_Opordered__ab__group__add: $i > $o ).

thf(class_Ring__and__Field_Oordered__semidom_decl,type,
    class_Ring__and__Field_Oordered__semidom: $i > $o ).

thf(class_Ring__and__Field_Opordered__semiring_decl,type,
    class_Ring__and__Field_Opordered__semiring: $i > $o ).

thf(class_Ring__and__Field_Oordered__idom_decl,type,
    class_Ring__and__Field_Oordered__idom: $i > $o ).

thf(class_Ring__and__Field_Ocomm__ring__1_decl,type,
    class_Ring__and__Field_Ocomm__ring__1: $i > $o ).

thf(class_Ring__and__Field_Oring__1__no__zero__divisors_decl,type,
    class_Ring__and__Field_Oring__1__no__zero__divisors: $i > $o ).

thf(class_OrderedGroup_Ocancel__ab__semigroup__add_decl,type,
    class_OrderedGroup_Ocancel__ab__semigroup__add: $i > $o ).

thf(class_Ring__and__Field_Oordered__ring__strict_decl,type,
    class_Ring__and__Field_Oordered__ring__strict: $i > $o ).

thf(class_Ring__and__Field_Opordered__ring_decl,type,
    class_Ring__and__Field_Opordered__ring: $i > $o ).

thf(class_Ring__and__Field_Odivision__ring_decl,type,
    class_Ring__and__Field_Odivision__ring: $i > $o ).

thf(c_Int_OMin_decl,type,
    c_Int_OMin: $i ).

thf(class_Ring__and__Field_Oring__1_decl,type,
    class_Ring__and__Field_Oring__1: $i > $o ).

thf(class_Ring__and__Field_Osemiring__1_decl,type,
    class_Ring__and__Field_Osemiring__1: $i > $o ).

thf(c_Int_Oiszero_decl,type,
    c_Int_Oiszero: $i > $i > $o ).

thf(class_OrderedGroup_Ocancel__semigroup__add_decl,type,
    class_OrderedGroup_Ocancel__semigroup__add: $i > $o ).

thf(class_Ring__and__Field_Oordered__semiring__strict_decl,type,
    class_Ring__and__Field_Oordered__semiring__strict: $i > $o ).

thf(c_Divides_Odiv__class_Omod_decl,type,
    c_Divides_Odiv__class_Omod: $i > $i > $i > $i ).

thf(c_Fact_Ofact__class_Ofact_decl,type,
    c_Fact_Ofact__class_Ofact: $i > $i > $i ).

thf(class_Ring__and__Field_Olordered__ring_decl,type,
    class_Ring__and__Field_Olordered__ring: $i > $o ).

thf(class_Ring__and__Field_Osemiring__0_decl,type,
    class_Ring__and__Field_Osemiring__0: $i > $o ).

thf(class_OrderedGroup_Omonoid__mult_decl,type,
    class_OrderedGroup_Omonoid__mult: $i > $o ).

thf(c_Parity_Oeven__odd__class_Oeven_decl,type,
    c_Parity_Oeven__odd__class_Oeven: $i > $i > $o ).

thf(class_OrderedGroup_Oab__group__add_decl,type,
    class_OrderedGroup_Oab__group__add: $i > $o ).

thf(class_OrderedGroup_Opordered__ab__semigroup__add_decl,type,
    class_OrderedGroup_Opordered__ab__semigroup__add: $i > $o ).

thf(class_Ring__and__Field_Oordered__comm__semiring__strict_decl,type,
    class_Ring__and__Field_Oordered__comm__semiring__strict: $i > $o ).

thf(class_Ring__and__Field_Omult__mono1_decl,type,
    class_Ring__and__Field_Omult__mono1: $i > $o ).

thf(class_Ring__and__Field_Osemiring_decl,type,
    class_Ring__and__Field_Osemiring: $i > $o ).

thf(class_Ring__and__Field_Oordered__semiring_decl,type,
    class_Ring__and__Field_Oordered__semiring: $i > $o ).

thf(class_OrderedGroup_Omonoid__add_decl,type,
    class_OrderedGroup_Omonoid__add: $i > $o ).

thf(c_Groebner__Basis_Ogb__semiring_decl,type,
    c_Groebner__Basis_Ogb__semiring: $i > $i > $i > $i > $i > $i > $o ).

thf(class_Ring__and__Field_Ozero__neq__one_decl,type,
    class_Ring__and__Field_Ozero__neq__one: $i > $o ).

thf(class_Ring__and__Field_Ocomm__semiring_decl,type,
    class_Ring__and__Field_Ocomm__semiring: $i > $o ).

thf(class_Ring__and__Field_Ono__zero__divisors_decl,type,
    class_Ring__and__Field_Ono__zero__divisors: $i > $o ).

thf(class_Ring__and__Field_Omult__zero_decl,type,
    class_Ring__and__Field_Omult__zero: $i > $o ).

thf(class_Power_Opower_decl,type,
    class_Power_Opower: $i > $o ).

thf(class_Int_Oring__char__0_decl,type,
    class_Int_Oring__char__0: $i > $o ).

thf(class_Divides_Oring__div_decl,type,
    class_Divides_Oring__div: $i > $o ).

thf(class_Ring__and__Field_Oring__no__zero__divisors_decl,type,
    class_Ring__and__Field_Oring__no__zero__divisors: $i > $o ).

thf(class_Ring__and__Field_Oring_decl,type,
    class_Ring__and__Field_Oring: $i > $o ).

thf(class_Ring__and__Field_Omult__mono_decl,type,
    class_Ring__and__Field_Omult__mono: $i > $o ).

thf(class_OrderedGroup_Opordered__cancel__ab__semigroup__add_decl,type,
    class_OrderedGroup_Opordered__cancel__ab__semigroup__add: $i > $o ).

thf(class_OrderedGroup_Ocomm__monoid__mult_decl,type,
    class_OrderedGroup_Ocomm__monoid__mult: $i > $o ).

thf(class_OrderedGroup_Oab__semigroup__add_decl,type,
    class_OrderedGroup_Oab__semigroup__add: $i > $o ).

thf(class_RealVector_Oreal__normed__algebra_decl,type,
    class_RealVector_Oreal__normed__algebra: $i > $o ).

thf(class_OrderedGroup_Oab__semigroup__idem__mult_decl,type,
    class_OrderedGroup_Oab__semigroup__idem__mult: $i > $o ).

thf(c_Fun_Oinj__on_decl,type,
    c_Fun_Oinj__on: $i > $i > $i > $i > $o ).

thf(c_Ring__and__Field_Ocomm__ring__1_decl,type,
    c_Ring__and__Field_Ocomm__ring__1: $i > $i > $i > $i > $i > $i > $i > $o ).

thf(c_COMBC_decl,type,
    c_COMBC: $i > $i > $i > $i > $i > $i > $i ).

thf(c_Parity_Osko__Parity__Xodd__Suc__mult__two__ex__1__1_decl,type,
    c_Parity_Osko__Parity__Xodd__Suc__mult__two__ex__1__1: $i > $i ).

thf(class_RealVector_Oreal__field_decl,type,
    class_RealVector_Oreal__field: $i > $o ).

thf(c_Complex_Oii_decl,type,
    c_Complex_Oii: $i ).

thf(277,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__0__iff_1) ).

thf(2452,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(1134,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_ODFT ) ) )
        = ( B @ ( A @ ( v_ka____ @ c_FFT__Mirabelle_OFFT ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc_Ohyps_0) ).

thf(4447,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_ODFT ) ) )
        = ( B @ ( A @ ( v_ka____ @ c_FFT__Mirabelle_OFFT ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).

thf(1607,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__if_0) ).

thf(5561,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1607]) ).

thf(1750,axiom,
    ! [A: $i] :
      ( c_Int_OPls
     != ( A @ c_Int_OBit1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I39_J_0) ).

thf(5896,plain,
    ! [A: $i] :
      ( c_Int_OPls
     != ( A @ c_Int_OBit1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1750]) ).

thf(1254,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( A @ ( B @ ( L @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__times_0) ).

thf(4729,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( A @ ( B @ ( L @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1254]) ).

thf(704,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__add_0) ).

thf(3434,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(461,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__imp__inverse__less_0) ).

thf(2876,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(1536,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__nat__equiv__def_1) ).

thf(5390,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1536]) ).

thf(710,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__of__pos_0) ).

thf(3449,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(1799,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__diff__Suc_0) ).

thf(5999,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1799]) ).

thf(914,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( L @ ( K @ ( A @ ( J @ ( B @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( G @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__le__even__power_H_0) ).

thf(3927,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( L @ ( K @ ( A @ ( J @ ( B @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( G @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(1315,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_8) ).

thf(4877,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1315]) ).

thf(191,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I23_J_0) ).

thf(2250,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(1715,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel__left__pos_1) ).

thf(5810,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1715]) ).

thf(1649,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__Suc__eq__Suc__mod_0) ).

thf(5659,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1649]) ).

thf(525,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__iff_1) ).

thf(3021,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(1556,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ab__semigroup__mult__class_Omult__ac_I1_J_0) ).

thf(5437,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1556]) ).

thf(436,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_0) ).

thf(2822,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(355,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__mult__iff_2) ).

thf(2630,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(889,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_Int_Oint @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__nat__number__of_0) ).

thf(3873,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_Int_Oint @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(1569,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( D @ ( F @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__diff__cong_0) ).

thf(5469,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( D @ ( F @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1569]) ).

thf(922,axiom,
    c_Int_OMin != c_Int_OPls,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I40_J_0) ).

thf(3944,plain,
    c_Int_OMin != c_Int_OPls,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(1526,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( L @ ( A @ ( K @ ( B @ ( H @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( H @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Onot__sum__power2__lt__zero_0) ).

thf(5365,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( L @ ( A @ ( K @ ( B @ ( H @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( H @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1526]) ).

thf(348,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__add__self2_0) ).

thf(2612,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(985,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( H @ ( G @ ( D @ ( F @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__ring__1_Opower__minus1__even_0) ).

thf(4094,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( H @ ( G @ ( D @ ( F @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(1406,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mult__le__cancel1_1) ).

thf(5088,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1406]) ).

thf(402,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__iff_5) ).

thf(2744,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(1299,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__cancel2_0) ).

thf(4835,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1299]) ).

thf(467,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__iff_1) ).

thf(2889,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(3,negated_conjecture,
    ( ( tc_Complex_Ocomplex @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( c_Suc @ c_COMBB ) ) ) ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_ODFT ) ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_ODFT ) ) @ c_HOL_Ominus__class_Ominus ) ) )
   != ( tc_Complex_Ocomplex @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( c_Suc @ c_COMBB ) ) ) ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ c_FFT__Mirabelle_OFFT ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ c_FFT__Mirabelle_OFFT ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_conjecture_2) ).

thf(1823,plain,
    ( ( tc_Complex_Ocomplex @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( c_Suc @ c_COMBB ) ) ) ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_ODFT ) ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_ODFT ) ) @ c_HOL_Ominus__class_Ominus ) ) )
   != ( tc_Complex_Ocomplex @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( c_Suc @ c_COMBB ) ) ) ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ c_FFT__Mirabelle_OFFT ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( tc_Complex_Ocomplex @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( v_a____ @ c_COMBB ) ) ) ) @ ( v_ka____ @ c_FFT__Mirabelle_OFFT ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(1135,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__diff__split_1) ).

thf(4450,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).

thf(230,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__comm__semiring__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oordered__comm__semiring__strict) ).

thf(2341,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(1187,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I25_J_0) ).

thf(4571,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1187]) ).

thf(540,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__power__minus__one_0) ).

thf(3053,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(228,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__eq__refl_0) ).

thf(2336,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(133,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__mult__distrib_0) ).

thf(2109,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(1620,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( tc_nat @ ( C @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__diff__rev__if_1) ).

thf(5593,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( tc_nat @ ( C @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1620]) ).

thf(97,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__mult__iff_3) ).

thf(2031,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(1056,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power_0) ).

thf(4262,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).

thf(1510,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__gt__zero__iff_2) ).

thf(5329,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1510]) ).

thf(798,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_1) ).

thf(3661,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(972,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__0_0) ).

thf(4062,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(689,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__triangle__ineq3_0) ).

thf(3399,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(81,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__minus__divide__right_0) ).

thf(1995,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(482,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_left__minus_0) ).

thf(2921,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(814,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Int_OBit1 )
       != ( B @ c_Int_OBit1 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I51_J_0) ).

thf(3698,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Int_OBit1 )
       != ( B @ c_Int_OBit1 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(937,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__add_2) ).

thf(3979,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(420,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_5) ).

thf(2787,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(1647,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__is__0_1) ).

thf(5654,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1647]) ).

thf(1656,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( A @ ( B @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__add__iff2_1) ).

thf(5676,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( A @ ( B @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1656]) ).

thf(340,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_3) ).

thf(2595,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(16,axiom,
    tc_Int_Oint @ class_OrderedGroup_Olordered__ab__group__add__abs,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Olordered__ab__group__add__abs) ).

thf(1850,plain,
    tc_Int_Oint @ class_OrderedGroup_Olordered__ab__group__add__abs,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(266,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__le__neq__trans_0) ).

thf(2427,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(75,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right2_5) ).

thf(1982,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(1815,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__irrefl_0) ).

thf(6042,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1815]) ).

thf(1234,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__add_0) ).

thf(4682,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1234]) ).

thf(759,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__0__iff_2) ).

thf(3566,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(1381,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__iff2_0) ).

thf(5028,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1381]) ).

thf(62,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq_1) ).

thf(1956,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(59,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__iff_2) ).

thf(1950,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(1366,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Oadd__mul__solve_1) ).

thf(4992,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1366]) ).

thf(204,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__frac__num_0) ).

thf(2280,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(1327,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( K @ ( E @ ( D @ ( J @ ( A @ ( I @ ( C @ ( B @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__gt__zero__iff_0) ).

thf(4904,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( K @ ( E @ ( D @ ( J @ ( A @ ( I @ ( C @ ( B @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1327]) ).

thf(942,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq__number__of_3) ).

thf(3990,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(549,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__minus__eq_0) ).

thf(3072,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(977,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( B = D )
      | ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ ( c_Suc @ hAPP ) @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__inject__base_0) ).

thf(4075,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( B = D )
      | ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ ( c_Suc @ hAPP ) @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(416,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Oadd__c_0) ).

thf(2776,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(1354,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__1__iff_1) ).

thf(4960,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1354]) ).

thf(1261,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq_0) ).

thf(4747,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1261]) ).

thf(1148,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__zero__eq__number__of_1) ).

thf(4479,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).

thf(1172,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( A @ ( E @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__0_0) ).

thf(4532,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( A @ ( E @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1172]) ).

thf(117,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oring__1) ).

thf(2077,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(963,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__0__iff_1) ).

thf(4043,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(746,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( D @ ( C @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( B @ ( A @ hAPP ) )
        = ( B @ ( D @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Oabs__sgn_0) ).

thf(3534,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( D @ ( C @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( B @ ( A @ hAPP ) )
        = ( B @ ( D @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(1563,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__diff__conv_0) ).

thf(5454,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1563]) ).

thf(597,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__cancel__end_1) ).

thf(3185,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(961,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( E @ ( A @ ( D @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I33_J_0) ).

thf(4037,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( E @ ( A @ ( D @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(505,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__strict__right__mono_0) ).

thf(2976,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(1104,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__imp__diff__less_0) ).

thf(4376,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).

thf(136,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__0_0) ).

thf(2117,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(131,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of_4) ).

thf(2104,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(1463,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__zero__eq__number__of_0) ).

thf(5225,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1463]) ).

thf(1439,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__int__code_I13_J_0) ).

thf(5168,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1439]) ).

thf(1040,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__number__of__Bit0_0) ).

thf(4226,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(1719,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__le__mult__iff_1) ).

thf(5820,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1719]) ).

thf(1088,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__Suc0_0) ).

thf(4339,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).

thf(904,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__if_1) ).

thf(3905,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(1164,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel__left__pos_0) ).

thf(4515,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1164]) ).

thf(862,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I16_J_0) ).

thf(3813,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(617,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__divide__distrib_0) ).

thf(3230,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(248,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( C @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__one__over_0) ).

thf(2381,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( C @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(1643,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__abs_0) ).

thf(5644,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1643]) ).

thf(476,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oring__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oring__no__zero__divisors) ).

thf(2908,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(1728,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__eq_0) ).

thf(5842,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1728]) ).

thf(111,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__assoc_0) ).

thf(2062,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(752,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( L @ ( C @ ( B @ ( K @ ( A @ ( J @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Opower2__less__0_0) ).

thf(3549,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( L @ ( C @ ( B @ ( K @ ( A @ ( J @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(736,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__diff__split_0) ).

thf(3509,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(1471,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__eq__square_0) ).

thf(5242,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1471]) ).

thf(678,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_4) ).

thf(3377,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(309,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__le__1__iff_0) ).

thf(2527,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(534,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_3) ).

thf(3040,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(583,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__less__iff_1) ).

thf(3153,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(494,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of_4) ).

thf(2953,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(338,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left1_5) ).

thf(2590,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(715,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ ( B @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__imp_0) ).

thf(3460,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ ( B @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(492,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__frac__eq_0) ).

thf(2947,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(1743,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__Suc1_0) ).

thf(5878,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1743]) ).

thf(560,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__divide__left_0) ).

thf(3098,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(373,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__inverse__eq_0) ).

thf(2671,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(375,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( B @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq_4) ).

thf(2676,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( B @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(90,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__divide_0) ).

thf(2014,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(906,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I15_J_0) ).

thf(3910,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(1200,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__left__cancel_0) ).

thf(4603,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).

thf(521,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__iff_2) ).

thf(3012,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(128,axiom,
    tc_Int_Oint @ class_OrderedGroup_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Ocancel__semigroup__add) ).

thf(2099,plain,
    tc_Int_Oint @ class_OrderedGroup_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(1462,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Odivision__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Odivision__ring) ).

thf(5224,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Odivision__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1462]) ).

thf(1653,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__sum__power2__lt__zero_0) ).

thf(5669,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1653]) ).

thf(1542,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__nat__plus__one__div__two_0) ).

thf(5405,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1542]) ).

thf(443,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide_Ozero_0) ).

thf(2837,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(1082,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__mono_0) ).

thf(4326,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).

thf(1481,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__iff__add_1) ).

thf(5265,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1481]) ).

thf(732,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__1_3) ).

thf(3498,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(1035,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel2_0) ).

thf(4215,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).

thf(1661,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__0__less__mult__iff_2) ).

thf(5691,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1661]) ).

thf(1204,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I16_J_1) ).

thf(4614,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1204]) ).

thf(450,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__le__1__iff_2) ).

thf(2854,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(1469,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__add__assoc2_0) ).

thf(5237,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1469]) ).

thf(1811,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone )
        = ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comp__arith_I110_J_0) ).

thf(6031,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone )
        = ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1811]) ).

thf(1592,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__div__eq__add3__div__number__of_0) ).

thf(5529,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1592]) ).

thf(1359,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__is__0__eq_1) ).

thf(4973,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1359]) ).

thf(612,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__1_0) ).

thf(3217,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(1755,axiom,
    tc_nat @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Divides_Osemiring__div) ).

thf(5906,plain,
    tc_nat @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1755]) ).

thf(1312,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( c_Suc @ hAPP ) )
        = B )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__lessI_0) ).

thf(4870,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( c_Suc @ hAPP ) )
        = B )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1312]) ).

thf(219,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I3_J_0) ).

thf(2316,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(1504,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_9) ).

thf(5316,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1504]) ).

thf(781,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__eq__cancel1_0) ).

thf(3616,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(811,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_termination__basic__simps_I4_J_0) ).

thf(3693,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(1436,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__add__iff1_1) ).

thf(5161,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1436]) ).

thf(1049,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Ocancel__ab__semigroup__add) ).

thf(4247,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(1362,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Oadd__r0__iff_0) ).

thf(4980,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1362]) ).

thf(821,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__field,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__RealVector_Oreal__normed__field) ).

thf(3717,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__field,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(1268,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comp__arith_I112_J_0) ).

thf(4762,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1268]) ).

thf(878,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__eq_2) ).

thf(3848,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(168,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Olordered__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Olordered__ring) ).

thf(2195,plain,
    tc_Int_Oint @ class_Ring__and__Field_Olordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(1783,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__imp__less__right_0) ).

thf(5967,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1783]) ).

thf(1551,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__mult__distrib_0) ).

thf(5425,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1551]) ).

thf(854,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__lt__two__imp__zero__or__one_0) ).

thf(3794,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(1212,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( F @ ( E @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( F @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__add_0) ).

thf(4633,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( F @ ( E @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( F @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1212]) ).

thf(983,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__le__add__iff2_1) ).

thf(4090,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(1524,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_8) ).

thf(5361,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1524]) ).

thf(538,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__0__left_0) ).

thf(3049,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(833,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_5) ).

thf(3745,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(1121,axiom,
    ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__eq__1__iff_2) ).

thf(4418,plain,
    ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).

thf(1258,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( C @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__inverse_0) ).

thf(4739,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( C @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1258]) ).

thf(346,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__mult__pos__le_1) ).

thf(2607,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(379,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__neq__zero_0) ).

thf(2686,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(517,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_5) ).

thf(3004,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(99,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__imp__le__div__pos_0) ).

thf(2035,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(1219,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod2__gr__0_1) ).

thf(4647,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1219]) ).

thf(951,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( L @ ( D @ ( C @ ( K @ ( A @ ( J @ ( I @ ( B @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( F = B )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__less__power2_1) ).

thf(4011,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( L @ ( D @ ( C @ ( K @ ( A @ ( J @ ( I @ ( B @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( F = B )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(574,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__1__iff_0) ).

thf(3135,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(986,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__1_0) ).

thf(4097,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(956,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__Suc__diff__eq2_0) ).

thf(4026,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(536,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__add__self1_0) ).

thf(3044,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(1491,axiom,
    tc_nat @ ( v_ka____ @ ( c_Suc @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Oord__class_Oless ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_i_0) ).

thf(5288,plain,
    tc_nat @ ( v_ka____ @ ( c_Suc @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1491]) ).

thf(991,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__diff__left_0) ).

thf(4110,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(1169,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__mult__distrib_0) ).

thf(4528,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).

thf(663,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__minus__iff_1) ).

thf(3338,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(487,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( C @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( B @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Rational_Oordered__idom_Osgn__greater_1) ).

thf(2933,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( C @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( B @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(1477,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__iff2_1) ).

thf(5257,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1477]) ).

thf(1214,axiom,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add__imp__le) ).

thf(4638,plain,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1214]) ).

thf(26,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__add_0) ).

thf(1872,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(849,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__n__not__le__n_0) ).

thf(3781,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(1131,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__eqI_1) ).

thf(4440,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).

thf(1220,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Oadd__mul__solve_1) ).

thf(4650,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1220]) ).

thf(1671,axiom,
    ! [A: $i] :
      ( A
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Oadd__r0__iff_1) ).

thf(5713,plain,
    ! [A: $i] :
      ( A
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1671]) ).

thf(288,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__abs__inverse_0) ).

thf(2476,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(354,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oinverse__class_Oinverse ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__inverse__eq_0) ).

thf(2627,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oinverse__class_Oinverse ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(157,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__left__commute_0) ).

thf(2168,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(653,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__mult2_0) ).

thf(3313,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(1701,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__less__imp__less_0) ).

thf(5782,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1701]) ).

thf(61,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__le__power_0) ).

thf(1954,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(446,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__not__less__zero_0) ).

thf(2844,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(1129,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__add__iff1_1) ).

thf(4435,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).

thf(102,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_1) ).

thf(2042,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(1545,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( H
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) )
      | ~ ( F @ ( L @ ( E @ ( D @ ( K @ ( J @ ( I @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__eq__zero__iff_1) ).

thf(5412,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( H
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) )
      | ~ ( F @ ( L @ ( E @ ( D @ ( K @ ( J @ ( I @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1545]) ).

thf(1817,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod2__gr__0_0) ).

thf(6046,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1817]) ).

thf(637,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__iff_2) ).

thf(3276,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(562,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__neq__one_0) ).

thf(3103,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(1788,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I28_J_0) ).

thf(5977,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1788]) ).

thf(1537,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__add__0_0) ).

thf(5393,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1537]) ).

thf(1686,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__nat__eq__number__of_1) ).

thf(5745,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1686]) ).

thf(1696,axiom,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat_Osimps_I3_J_0) ).

thf(5770,plain,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1696]) ).

thf(958,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__eq_1) ).

thf(4030,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(159,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__mono__nat_0) ).

thf(2173,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(1594,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower3__eq__cube_0) ).

thf(5533,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1594]) ).

thf(1280,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__antisym_0) ).

thf(4792,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1280]) ).

thf(8,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__eq__0_1) ).

thf(1835,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(178,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_lessequals ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__0__eq_0) ).

thf(2216,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_lessequals ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(619,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__neq__le__trans_0) ).

thf(3236,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(233,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__nonzero__nat_0) ).

thf(2348,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(1464,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__less__mono_0) ).

thf(5227,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1464]) ).

thf(503,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__0__le__iff__le_1) ).

thf(2972,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(104,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B = C )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__inverse__eq__imp__eq_0) ).

thf(2046,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B = C )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(509,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide_Ominus_0) ).

thf(2985,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(1156,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( D @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_left__add__mult__distrib_0) ).

thf(4496,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( D @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).

thf(1588,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__leE_0) ).

thf(5518,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1588]) ).

thf(1058,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__eq__if_1) ).

thf(4266,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(1206,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__refl_0) ).

thf(4618,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1206]) ).

thf(556,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__1__iff_0) ).

thf(3088,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(651,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__zero__left_0) ).

thf(3308,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(87,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oring__1__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oring__1__no__zero__divisors) ).

thf(2010,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(923,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__Suc__eq_2) ).

thf(3947,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(996,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_5) ).

thf(4124,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(466,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__1__right_0) ).

thf(2886,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(1753,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__self4_0) ).

thf(5903,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1753]) ).

thf(639,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__minus1_0) ).

thf(3281,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(717,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left1_0) ).

thf(3466,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(1267,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__trans_0) ).

thf(4760,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1267]) ).

thf(1355,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Ozero__left_0) ).

thf(4963,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1355]) ).

thf(1009,axiom,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_numeral__3__eq__3_0) ).

thf(4156,plain,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).

thf(1483,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__le__mult__iff_2) ).

thf(5269,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1483]) ).

thf(892,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__mult__distrib2_0) ).

thf(3880,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(891,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__strict__increasing_0) ).

thf(3878,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(1688,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__2__right_0) ).

thf(5750,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1688]) ).

thf(1348,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__product__nat_2) ).

thf(4948,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1348]) ).

thf(311,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__idom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oordered__idom) ).

thf(2531,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__idom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(1143,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq__number__of_2) ).

thf(4467,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).

thf(1520,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_8) ).

thf(5352,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1520]) ).

thf(1516,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__add__assoc_0) ).

thf(5342,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1516]) ).

thf(632,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__minus__eq_0) ).

thf(3263,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(384,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__le__zero__iff_0) ).

thf(2698,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(1238,axiom,
    ! [A: $i] :
      ( ( A @ c_Int_OBit0 )
     != c_Int_OMin ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I45_J_0) ).

thf(4692,plain,
    ! [A: $i] :
      ( ( A @ c_Int_OBit0 )
     != c_Int_OMin ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1238]) ).

thf(1692,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__lessD_0) ).

thf(5759,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1692]) ).

thf(762,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( A @ ( B @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__diff__less__iff_0) ).

thf(3574,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( A @ ( B @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(543,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__eq__neg_1) ).

thf(3061,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(388,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( B @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( D @ ( E @ ( D @ ( C @ ( A @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( C @ ( A @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__commutes_0) ).

thf(2708,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( B @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( D @ ( E @ ( D @ ( C @ ( A @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( C @ ( A @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(298,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right1_3) ).

thf(2501,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(1086,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__one__even__power_0) ).

thf(4334,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).

thf(779,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq__number__of_2) ).

thf(3612,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(726,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__iff_1) ).

thf(3485,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(847,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__le__cancel1_0) ).

thf(3776,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(164,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( C @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq_0) ).

thf(2185,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( C @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(819,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__eq__number__of_2) ).

thf(3713,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(865,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I36_J_0) ).

thf(3820,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(898,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Oabs__mult_0) ).

thf(3891,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(625,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_9) ).

thf(3248,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(606,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_8) ).

thf(3204,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(1775,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring__0) ).

thf(5953,plain,
    tc_nat @ class_Ring__and__Field_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1775]) ).

thf(707,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__self1__is__id_0) ).

thf(3442,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(936,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nat__number__of_1) ).

thf(3976,plain,
    ! [B: $i,A: $i] :
      ( ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(225,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right2_2) ).

thf(2328,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(602,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__1__iff_0) ).

thf(3195,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(529,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__div__equality2_0) ).

thf(3029,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(1500,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I14_J_0) ).

thf(5306,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1500]) ).

thf(1323,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__less__cancel__left_0) ).

thf(4896,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1323]) ).

thf(1812,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_0) ).

thf(6034,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1812]) ).

thf(1814,axiom,
    ! [A: $i] :
      ( ( c_Int_OMin = A )
      | ( c_Int_OMin
       != ( A @ c_Int_OBit1 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I43_J_0) ).

thf(6039,plain,
    ! [A: $i] :
      ( ( c_Int_OMin = A )
      | ( c_Int_OMin
       != ( A @ c_Int_OBit1 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1814]) ).

thf(975,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__frac__eq_0) ).

thf(4070,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(371,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left1_3) ).

thf(2667,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(1007,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__add__iff2_1) ).

thf(4151,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(406,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( I @ ( C @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( K @ ( H @ ( J @ ( I @ ( H @ hAPP ) @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__1__pos_0) ).

thf(2755,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( I @ ( C @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( K @ ( H @ ( J @ ( I @ ( H @ hAPP ) @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(1757,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__product__nat_0) ).

thf(5910,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1757]) ).

thf(377,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__imp__neg__le_0) ).

thf(2682,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(570,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_2) ).

thf(3125,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(4,axiom,
    tc_Int_Oint @ class_OrderedGroup_Oordered__ab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Oordered__ab__group__add) ).

thf(1826,plain,
    tc_Int_Oint @ class_OrderedGroup_Oordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(835,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__0__le__power__imp__0__le_0) ).

thf(3749,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(327,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_xt1_I12_J_0) ).

thf(2562,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(955,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__minus1__right_0) ).

thf(4023,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(7,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__equal__zero_1) ).

thf(1832,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(1794,axiom,
    tc_nat @ class_OrderedGroup_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Omonoid__mult) ).

thf(5989,plain,
    tc_nat @ class_OrderedGroup_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1794]) ).

thf(80,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__eqI_0) ).

thf(1992,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(1492,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel1_2) ).

thf(5289,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1492]) ).

thf(1668,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq_5) ).

thf(5706,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1668]) ).

thf(1186,axiom,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
     != A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__n__not__n_0) ).

thf(4568,plain,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).

thf(669,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( G
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( E @ ( L @ ( D @ ( C @ ( F @ ( A @ ( K @ ( J @ ( B @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( B @ ( F @ hAPP ) @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__less__power__abs__iff_0) ).

thf(3353,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( G
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( E @ ( L @ ( D @ ( C @ ( F @ ( A @ ( K @ ( J @ ( B @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( B @ ( F @ hAPP ) @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(858,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__neqE__nat_0) ).

thf(3803,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(1415,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( K @ ( A @ ( J @ ( B @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( C @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Oodd__0__le__power__imp__0__le_0) ).

thf(5108,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( K @ ( A @ ( J @ ( B @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( C @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1415]) ).

thf(1488,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ab__diff__minus_0) ).

thf(5279,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1488]) ).

thf(1566,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_5) ).

thf(5461,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1566]) ).

thf(386,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__inverse__eq__divide_0) ).

thf(2703,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(1685,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Oidom) ).

thf(5744,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1685]) ).

thf(47,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Oadd__cancel_0) ).

thf(1922,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(789,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__diff__conv_1) ).

thf(3638,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(1710,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__power__nat_2) ).

thf(5802,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1710]) ).

thf(1573,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__is__add_5) ).

thf(5480,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1573]) ).

thf(554,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_pos__add__strict_0) ).

thf(3084,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(925,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq__number__of_0) ).

thf(3951,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(1402,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__number__of_0) ).

thf(5076,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1402]) ).

thf(886,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_0) ).

thf(3866,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(1631,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ofield,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Ofield) ).

thf(5617,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ofield,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1631]) ).

thf(1227,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__strict__mono_0) ).

thf(4669,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1227]) ).

thf(1818,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Omul__a_0) ).

thf(6049,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1818]) ).

thf(1514,axiom,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__gt__zero__nat_0) ).

thf(5338,plain,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1514]) ).

thf(390,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( E @ c_lessequals ) ) )
      | ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_frac__less_0) ).

thf(2713,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( E @ c_lessequals ) ) )
      | ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(945,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__le__add__iff1_1) ).

thf(3998,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(874,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Oring__no__zero__divisors) ).

thf(3840,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(716,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__positive__imp__positive_0) ).

thf(3463,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(677,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I22_J_0) ).

thf(3374,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(682,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__0__iff_3) ).

thf(3385,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(452,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__nonpos__nonpos_0) ).

thf(2857,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(1738,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__eq__cancel__disj_0) ).

thf(5866,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1738]) ).

thf(1309,axiom,
    ! [A: $i] :
      ( ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__exhaust__less__4_0) ).

thf(4862,plain,
    ! [A: $i] :
      ( ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1309]) ).

thf(1096,axiom,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_numeral__2__eq__2_0) ).

thf(4358,plain,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).

thf(105,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_0) ).

thf(2049,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(322,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__one_0) ).

thf(2552,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(1225,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__Suc_1) ).

thf(4664,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1225]) ).

thf(412,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__iff_4) ).

thf(2767,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(1449,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_root__cancel_0) ).

thf(5193,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1449]) ).

thf(1625,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__assoc_0) ).

thf(5604,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1625]) ).

thf(1208,axiom,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__eq__power2_1) ).

thf(4622,plain,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1208]) ).

thf(910,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Oab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Oab__group__add) ).

thf(3919,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(1507,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right_Ominus_0) ).

thf(5323,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1507]) ).

thf(1338,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__cancel_0) ).

thf(4928,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1338]) ).

thf(1392,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__add_3) ).

thf(5052,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1392]) ).

thf(485,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_7) ).

thf(2929,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(1230,axiom,
    tc_nat @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Orderings_Opreorder) ).

thf(4675,plain,
    tc_nat @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1230]) ).

thf(719,axiom,
    tc_Int_Oint @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Power_Opower) ).

thf(3470,plain,
    tc_Int_Oint @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(1732,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__mono1_0) ).

thf(5851,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1732]) ).

thf(902,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__diff__iff_1) ).

thf(3902,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(829,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__mult__pos_0) ).

thf(3734,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(522,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_6) ).

thf(3014,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(1003,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I22_J_0) ).

thf(4141,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(94,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_3) ).

thf(2024,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(872,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__ring__1__class_Opower__minus__even_0) ).

thf(3835,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(817,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq_3) ).

thf(3708,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(1107,axiom,
    tc_Complex_Ocomplex @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Power_Opower) ).

thf(4383,plain,
    tc_Complex_Ocomplex @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).

thf(213,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_1) ).

thf(2301,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(1140,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__Suc__eq__div__add3_0) ).

thf(4460,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1140]) ).

thf(595,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of_2) ).

thf(3180,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(1110,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__0__left__number__of_1) ).

thf(4390,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).

thf(1161,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__le_2) ).

thf(4507,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1161]) ).

thf(27,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__cancel__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Opordered__cancel__semiring) ).

thf(1875,plain,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(449,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__pos__pos_0) ).

thf(2852,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(1272,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__div__cancel__disj_1) ).

thf(4771,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1272]) ).

thf(1052,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__Suc_0) ).

thf(4253,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(950,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_left__diff__distrib__number__of_0) ).

thf(4008,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(1786,axiom,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__Suc0_1) ).

thf(5974,plain,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1786]) ).

thf(1705,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_iszero__number__of__Bit1_0) ).

thf(5792,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1705]) ).

thf(1128,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__add__iff2_0) ).

thf(4432,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).

thf(1800,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__nat__plus__one__div__two_0) ).

thf(6002,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1800]) ).

thf(1112,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__self1__is__m_0) ).

thf(4396,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).

thf(423,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__0__less__iff__less_0) ).

thf(2794,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(285,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__imp__inverse__le_0) ).

thf(2469,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(802,axiom,
    ! [A: $i] :
      ( c_Int_OMin
     != ( A @ c_Int_OBit0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I42_J_0) ).

thf(3670,plain,
    ! [A: $i] :
      ( c_Int_OMin
     != ( A @ c_Int_OBit0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(1289,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__trans__Suc_0) ).

thf(4815,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1289]) ).

thf(209,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( D @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != C )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__cancel__end_0) ).

thf(2291,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( D @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != C )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(91,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__le__imp__le__neg_0) ).

thf(2017,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(161,axiom,
    tc_Int_Oint @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Orderings_Opreorder) ).

thf(2178,plain,
    tc_Int_Oint @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(1658,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Int_OBit0 )
       != ( B @ c_Int_OBit0 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I48_J_0) ).

thf(5682,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Int_OBit0 )
       != ( B @ c_Int_OBit0 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1658]) ).

thf(1703,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__Suc__1_0) ).

thf(5787,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1703]) ).

thf(1118,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__minus_0) ).

thf(4411,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).

thf(493,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__inverse_0) ).

thf(2950,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(1240,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I4_J_0) ).

thf(4696,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1240]) ).

thf(186,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I20_J_0) ).

thf(2237,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(1084,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__less__le__mono_0) ).

thf(4329,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).

thf(896,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__power__Suc_0) ).

thf(3887,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(604,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__less__imp__less__neg_0) ).

thf(3199,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(1747,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__numeral__1_0) ).

thf(5888,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1747]) ).

thf(1591,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__imp__eq_0) ).

thf(5526,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1591]) ).

thf(1132,axiom,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__number__of__Pls_0) ).

thf(4443,plain,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).

thf(550,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right1_0) ).

thf(3075,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(1622,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__zero__eq_1) ).

thf(5597,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1622]) ).

thf(123,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__less__0__iff__less_0) ).

thf(2090,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(687,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__add__abs_0) ).

thf(3393,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(1307,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__2__right_0) ).

thf(4857,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1307]) ).

thf(825,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right_Odiff_0) ).

thf(3724,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(1292,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_1) ).

thf(4820,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1292]) ).

thf(1734,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left_Odiff_0) ).

thf(5856,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1734]) ).

thf(743,axiom,
    tc_nat @ class_Ring__and__Field_Opordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Opordered__semiring) ).

thf(3527,plain,
    tc_nat @ class_Ring__and__Field_Opordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

thf(721,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__eq__zero__iff_1) ).

thf(3473,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(1772,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_5) ).

thf(5946,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1772]) ).

thf(967,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diffs0__imp__equal_0) ).

thf(4050,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(138,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq_3) ).

thf(2122,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(742,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBB ) ) ) ) @ hAPP ) )
      = ( F @ ( B @ hAPP ) @ ( A @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ATP__Linkup_OCOMBB__def_0) ).

thf(3525,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBB ) ) ) ) @ hAPP ) )
      = ( F @ ( B @ hAPP ) @ ( A @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(22,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__less__eq__nonneg_1) ).

thf(1864,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(935,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__sum_0) ).

thf(3973,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(1711,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ogroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Ogroup__add) ).

thf(5804,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1711]) ).

thf(73,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right1_0) ).

thf(1978,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(1597,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__mult__two__ex_1) ).

thf(5541,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1597]) ).

thf(1759,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__diff_0) ).

thf(5914,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1759]) ).

thf(1383,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__difference__nat_1) ).

thf(5032,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1383]) ).

thf(1434,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( C
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq_0) ).

thf(5156,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( C
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1434]) ).

thf(1820,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__pos__neg2_0) ).

thf(6054,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1820]) ).

thf(760,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I18_J_0) ).

thf(3569,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(1712,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__1__mult_0) ).

thf(5805,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1712]) ).

thf(1791,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__field,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__RealVector_Oreal__field) ).

thf(5985,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__field,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1791]) ).

thf(912,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_xt1_I10_J_0) ).

thf(3923,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(647,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_9) ).

thf(3300,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(558,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Oadd__0_0) ).

thf(3093,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(993,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Omul__a_0) ).

thf(4115,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(1693,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__diff__assoc_0) ).

thf(5761,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1693]) ).

thf(1182,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I13_J_0) ).

thf(4556,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).

thf(1478,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__self3_0) ).

thf(5259,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1478]) ).

thf(115,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__ge__minus__self_0) ).

thf(2073,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(1012,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__pred_0) ).

thf(4162,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).

thf(880,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_Int_Oint @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = A )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zless__linear_0) ).

thf(3851,plain,
    ! [B: $i,A: $i] :
      ( ( tc_Int_Oint @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = A )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(1263,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__strict__left__mono_0) ).

thf(4751,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1263]) ).

thf(969,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_6) ).

thf(4055,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(1457,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( F @ ( E @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__mult_0) ).

thf(5210,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( F @ ( E @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1457]) ).

thf(1412,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__add_0) ).

thf(5101,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1412]) ).

thf(32,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I24_J_0) ).

thf(1886,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(1506,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__iff__gr__or__eq_1) ).

thf(5321,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1506]) ).

thf(1336,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Ocomm__semiring__1) ).

thf(4924,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1336]) ).

thf(1294,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__Suc2_0) ).

thf(4825,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1294]) ).

thf(1242,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__gr__0_2) ).

thf(4701,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1242]) ).

thf(130,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__semiring__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oordered__semiring__strict) ).

thf(2103,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(410,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of1_3) ).

thf(2762,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(684,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left2_1) ).

thf(3389,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(1612,axiom,
    tc_nat @ class_OrderedGroup_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Omonoid__add) ).

thf(5572,plain,
    tc_nat @ class_OrderedGroup_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1612]) ).

thf(301,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_3) ).

thf(2508,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(1395,axiom,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
      = ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__eq__plus1__left_0) ).

thf(5059,plain,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
      = ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1395]) ).

thf(940,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__mono2_0) ).

thf(3986,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(109,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_3) ).

thf(2058,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(241,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_2) ).

thf(2367,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(747,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__iff_2) ).

thf(3537,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(49,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__half__sum_0) ).

thf(1928,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(1697,axiom,
    tc_nat @ class_OrderedGroup_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Oab__semigroup__add) ).

thf(5773,plain,
    tc_nat @ class_OrderedGroup_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1697]) ).

thf(441,axiom,
    tc_Int_Oint @ class_Int_Oring__char__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Int_Oring__char__0) ).

thf(2833,plain,
    tc_Int_Oint @ class_Int_Oring__char__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(1540,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__iff2_0) ).

thf(5401,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1540]) ).

thf(599,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__not__le_1) ).

thf(3189,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(1532,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ c_Int_OBit0 )
     != ( B @ c_Int_OBit1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I49_J_0) ).

thf(5379,plain,
    ! [B: $i,A: $i] :
      ( ( A @ c_Int_OBit0 )
     != ( B @ c_Int_OBit1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1532]) ).

thf(831,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( D @ ( D @ ( B @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower3__eq__cube_0) ).

thf(3739,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( D @ ( D @ ( B @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(166,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_5) ).

thf(2190,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(698,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of1_0) ).

thf(3421,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(1530,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ ( c_Suc @ hAPP ) )
       != ( B @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat_Oinject_0) ).

thf(5375,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ ( c_Suc @ hAPP ) )
       != ( B @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1530]) ).

thf(1534,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Zero__neq__Suc_0) ).

thf(5384,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1534]) ).

thf(860,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( E @ ( c_Suc @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__Suc2_0) ).

thf(3808,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( E @ ( c_Suc @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(307,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__1__iff_1) ).

thf(2523,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(532,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__le__cancel__right_1) ).

thf(3036,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(1099,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__number__of__add__1_0) ).

thf(4364,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).

thf(982,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = C )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__unique_0) ).

thf(4087,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = C )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(1368,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mult__cancel1_0) ).

thf(4996,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1368]) ).

thf(1495,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__mult__neg__le_1) ).

thf(5296,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1495]) ).

thf(414,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__imp__inverse__less__neg_0) ).

thf(2772,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(585,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__eq__neg__nonpos_1) ).

thf(3157,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(962,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__mod_1) ).

thf(4040,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

thf(672,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of_3) ).

thf(3361,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(838,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__square__less__zero_0) ).

thf(3756,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(1807,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__diff__cancel_0) ).

thf(6021,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1807]) ).

thf(430,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__le__dividend_0) ).

thf(2809,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(188,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add__abs )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__if__lattice_0) ).

thf(2242,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add__abs )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(359,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__diff__eq_0) ).

thf(2639,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(1217,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__not__refl_0) ).

thf(4643,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1217]) ).

thf(56,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__iff__diff__le__0_0) ).

thf(1943,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(499,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_2) ).

thf(2965,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(1637,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I12_J_0) ).

thf(5629,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1637]) ).

thf(1166,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq__number__of_0) ).

thf(4519,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1166]) ).

thf(1427,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__2_0) ).

thf(5137,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1427]) ).

thf(1245,axiom,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
        = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_lemma__Suc__Suc__4n__diff__2_0) ).

thf(4707,plain,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
        = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1245]) ).

thf(888,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__c_0) ).

thf(3870,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(777,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( E @ ( K @ ( D @ ( C @ ( J @ ( I @ ( H @ ( G @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__eq__power2_0) ).

thf(3606,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( E @ ( K @ ( D @ ( C @ ( J @ ( I @ ( H @ ( G @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(1721,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) )
       != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__Suc_0) ).

thf(5824,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) )
       != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1721]) ).

thf(1509,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__Suc__ex__iff_1) ).

thf(5328,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1509]) ).

thf(1033,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__le__mono_0) ).

thf(4211,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(392,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of_0) ).

thf(2718,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(173,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_9) ).

thf(2205,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(336,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__iff_5) ).

thf(2586,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(465,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__eq__neg__nonpos_0) ).

thf(2884,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(980,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__Suc__0_0) ).

thf(4082,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(973,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__le__zero__iff_2) ).

thf(4065,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(1663,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq_1) ).

thf(5695,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1663]) ).

thf(1496,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__iff2_1) ).

thf(5298,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1496]) ).

thf(901,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ c_Int_OBit1 )
     != ( B @ c_Int_OBit0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I50_J_0) ).

thf(3899,plain,
    ! [B: $i,A: $i] :
      ( ( A @ c_Int_OBit1 )
     != ( B @ c_Int_OBit0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(882,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__1_0) ).

thf(3857,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(1264,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel1_0) ).

thf(4753,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1264]) ).

thf(939,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__strict__right__mono__neg_0) ).

thf(3984,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(1171,axiom,
    tc_Complex_Ocomplex @ class_Int_Onumber__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Int_Onumber__ring) ).

thf(4531,plain,
    tc_Complex_Ocomplex @ class_Int_Onumber__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1171]) ).

thf(1429,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__mult2_0) ).

thf(5142,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1429]) ).

thf(44,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__le__cancel__right_0) ).

thf(1915,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(806,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I27_J_0) ).

thf(3679,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(582,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_3) ).

thf(3151,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(88,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__nonpos__neg_0) ).

thf(2011,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(1374,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__Suc_0) ).

thf(5012,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1374]) ).

thf(1742,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Omul__0_0) ).

thf(5875,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1742]) ).

thf(1322,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__diff__distrib__number__of_0) ).

thf(4893,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1322]) ).

thf(1081,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__one__even__power_0) ).

thf(4323,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).

thf(770,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__mult__pos2_0) ).

thf(3591,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(1194,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__difference__nat_0) ).

thf(4589,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).

thf(1116,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__neg__pos_0) ).

thf(4407,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).

thf(1805,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__diff__right__eq_0) ).

thf(6015,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1805]) ).

thf(823,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__eq__0__iff_1) ).

thf(3720,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(745,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__eq__zero__iff_1) ).

thf(3531,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(788,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__eq__1__iff_0) ).

thf(3635,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(1558,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_1) ).

thf(5443,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1558]) ).

thf(239,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__less__iff__less_0) ).

thf(2364,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(155,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( B @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left2_0) ).

thf(2164,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( B @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(1599,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__0_2) ).

thf(5545,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1599]) ).

thf(1400,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__a_0) ).

thf(5070,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1400]) ).

thf(1493,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Opwr__pwr_0) ).

thf(5291,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1493]) ).

thf(1384,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I12_J_0) ).

thf(5034,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1384]) ).

thf(944,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__diff__rev__if_0) ).

thf(3995,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(1419,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_5) ).

thf(5117,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1419]) ).

thf(655,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__less__iff_1) ).

thf(3318,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(1736,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__difference__nat_4) ).

thf(5862,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1736]) ).

thf(843,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gr__implies__not0_0) ).

thf(3767,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(1189,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__nat__number__of_0) ).

thf(4576,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).

thf(216,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__Numeral0_0) ).

thf(2307,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(120,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__le__D2_0) ).

thf(2083,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(296,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__imp__inverse__le__neg_0) ).

thf(2496,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(352,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__self2__is__id_0) ).

thf(2621,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(1076,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__Suc__eq__diff__pred_0) ).

thf(4312,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(368,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right2_4) ).

thf(2660,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(971,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__div__cancel1_0) ).

thf(4059,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(474,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__le__iff_1) ).

thf(2905,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(1512,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__0__eq__0_0) ).

thf(5334,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1512]) ).

thf(24,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_0) ).

thf(1868,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(1016,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel2_1) ).

thf(4172,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(1369,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__mult__distrib2_0) ).

thf(4999,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1369]) ).

thf(1421,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__gr__0_0) ).

thf(5122,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1421]) ).

thf(1337,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__even__abs__number__of_0) ).

thf(4925,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1337]) ).

thf(1455,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__less__cancel__right_1) ).

thf(5206,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1455]) ).

thf(1390,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__Suc__shift_0) ).

thf(5049,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1390]) ).

thf(38,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__zero_0) ).

thf(1900,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(1247,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( C = E )
      | ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Oadd__mul__solve_0) ).

thf(4712,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( C = E )
      | ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1247]) ).

thf(868,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__commute_0) ).

thf(3827,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(317,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__double__add__iff__zero__less__single__add_1) ).

thf(2541,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(1706,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__cancel2_1) ).

thf(5794,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1706]) ).

thf(1615,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_termination__basic__simps_I1_J_0) ).

thf(5579,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1615]) ).

thf(1674,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left__le__imp__le_0) ).

thf(5719,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1674]) ).

thf(1405,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__minus__eq__add_0) ).

thf(5085,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1405]) ).

thf(1690,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I29_J_0) ).

thf(5754,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1690]) ).

thf(1749,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Omul__d_0) ).

thf(5893,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1749]) ).

thf(1011,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__iff1_1) ).

thf(4160,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).

thf(1332,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I10_J_1) ).

thf(4915,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1332]) ).

thf(259,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__le__less__trans_0) ).

thf(2409,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(1628,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__semiring) ).

thf(5611,plain,
    tc_nat @ class_Ring__and__Field_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1628]) ).

thf(1681,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_xt1_I9_J_0) ).

thf(5734,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1681]) ).

thf(326,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left2_2) ).

thf(2560,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(428,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oidom) ).

thf(2805,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(1365,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power2_1) ).

thf(4989,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1365]) ).

thf(1388,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__less__cancel__right_0) ).

thf(5044,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1388]) ).

thf(1519,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__eqI_0) ).

thf(5349,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1519]) ).

thf(148,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_5) ).

thf(2147,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(1482,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__imp__diff__is__add_0) ).

thf(5266,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1482]) ).

thf(1777,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ab__semigroup__add__class_Oadd__ac_I1_J_0) ).

thf(5956,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1777]) ).

thf(1717,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Omul__c_0) ).

thf(5815,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1717]) ).

thf(832,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__Suc_0) ).

thf(3742,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(1090,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__diff__iff_0) ).

thf(4344,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).

thf(998,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( C = E )
      | ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Onoteq__reduce_0) ).

thf(4128,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( C = E )
      | ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(620,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Osemiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Osemiring__1) ).

thf(3239,plain,
    tc_Int_Oint @ class_Ring__and__Field_Osemiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(618,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__mult__distrib_0) ).

thf(3233,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(894,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__eq__number__of_0) ).

thf(3884,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(142,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__add__cancel_0) ).

thf(2132,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(261,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oinverse__class_Oinverse ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__inverse__inverse__eq_0) ).

thf(2414,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oinverse__class_Oinverse ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(626,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__square_0) ).

thf(3250,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(254,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__antisym__conv_0) ).

thf(2396,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(738,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__zero__eq__number__of_2) ).

thf(3514,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(807,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ c_Int_OBit1 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I9_J_1) ).

thf(3682,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ c_Int_OBit1 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(848,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I15_J_0) ).

thf(3778,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(897,axiom,
    ! [A: $i] :
      ~ ( tc_Int_Oint @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zless__le_1) ).

thf(3889,plain,
    ! [A: $i] :
      ~ ( tc_Int_Oint @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(720,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__less__iff_0) ).

thf(3471,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(1763,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__self__implies__10_0) ).

thf(5924,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1763]) ).

thf(641,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__less__eq__dividend_0) ).

thf(3286,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(31,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_equation__minus__iff_0) ).

thf(1883,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(1790,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__diff__conv_1) ).

thf(5983,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1790]) ).

thf(1024,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( B @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Omult__1__left_0) ).

thf(4192,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( B @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).

thf(822,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__le__eq_0) ).

thf(3718,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(176,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__power__minus_0) ).

thf(2211,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(365,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__triangle__ineq_0) ).

thf(2653,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(496,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__cancel__21_0) ).

thf(2958,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(924,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__one_0) ).

thf(3948,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(589,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__increasing_0) ).

thf(3166,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(274,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__nonnegative__iff__nonnegative_1) ).

thf(2445,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(1069,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__less__Suc__eq_0) ).

thf(4295,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(600,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__not__less_0) ).

thf(3191,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(54,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left2_4) ).

thf(1938,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(1064,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I9_J_0) ).

thf(4280,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).

thf(1744,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( B @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Omul__pwr_0) ).

thf(5879,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( B @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1744]) ).

thf(1666,axiom,
    ( ( c_Int_OPls @ c_Int_OBit0 )
    = c_Int_OPls ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Bit0__Pls_0) ).

thf(5702,plain,
    ( ( c_Int_OPls @ c_Int_OBit0 )
    = c_Int_OPls ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1666]) ).

thf(1498,axiom,
    tc_nat @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Orderings_Olinorder) ).

thf(5302,plain,
    tc_nat @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1498]) ).

thf(1578,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__eqI_1) ).

thf(5492,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1578]) ).

thf(1768,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Olordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__le__mult_0) ).

thf(5937,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Olordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1768]) ).

thf(634,axiom,
    tc_Int_Oint @ class_OrderedGroup_Oab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Oab__group__add) ).

thf(3269,plain,
    tc_Int_Oint @ class_OrderedGroup_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(1202,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( E @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I1_J_0) ).

thf(4608,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( E @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).

thf(46,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__antisym__conv1_1) ).

thf(1920,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(968,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__le__add__iff1_0) ).

thf(4053,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(268,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nonpos__neg_0) ).

thf(2433,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(1252,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__int__code_I13_J_1) ).

thf(4725,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1252]) ).

thf(68,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__le__mono_0) ).

thf(1968,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(557,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__eqI_1) ).

thf(3090,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(328,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( M @ ( C @ ( B @ ( A @ ( L @ ( K @ ( J @ ( I @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( F @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( F @ ( E @ ( A @ hAPP ) @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Opower__abs_0) ).

thf(2565,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( M @ ( C @ ( B @ ( A @ ( L @ ( K @ ( J @ ( I @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( F @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( F @ ( E @ ( A @ hAPP ) @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(438,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__le__inverse__iff_0) ).

thf(2826,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(435,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__less__iff_2) ).

thf(2820,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(1347,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ocomm__ring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Ocomm__ring__1) ).

thf(4947,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1347]) ).

thf(575,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__iff_3) ).

thf(3137,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(545,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Omult__mono,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Omult__mono) ).

thf(3065,plain,
    tc_Int_Oint @ class_Ring__and__Field_Omult__mono,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(1443,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I33_J_0) ).

thf(5177,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1443]) ).

thf(1367,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I17_J_1) ).

thf(4994,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1367]) ).

thf(646,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__commute_0) ).

thf(3297,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(459,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left1_5) ).

thf(2872,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(1378,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__less__mult_0) ).

thf(5021,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1378]) ).

thf(1793,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Ocomm__monoid__add) ).

thf(5988,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1793]) ).

thf(899,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__nonneg__nonneg_0) ).

thf(3894,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(113,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I21_J_0) ).

thf(2068,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(171,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__div__equality_0) ).

thf(2200,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(347,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( A @ ( K @ ( C @ ( J @ ( D @ ( I @ ( H @ ( E @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( ( B @ ( A @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__pos_0) ).

thf(2609,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( A @ ( K @ ( C @ ( J @ ( D @ ( I @ ( H @ ( E @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( ( B @ ( A @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(1022,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__less__mono__nat_0) ).

thf(4187,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).

thf(162,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq_0) ).

thf(2179,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(1602,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Oring) ).

thf(5551,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1602]) ).

thf(751,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__iff__gr__or__eq_0) ).

thf(3546,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(1386,axiom,
    ! [A: $i] :
      ( ( c_Int_OPls = A )
      | ( c_Int_OPls
       != ( A @ c_Int_OBit0 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I38_J_0) ).

thf(5038,plain,
    ! [A: $i] :
      ( ( c_Int_OPls = A )
      | ( c_Int_OPls
       != ( A @ c_Int_OBit0 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1386]) ).

thf(1114,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B = D )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__imp__eq__base_0) ).

thf(4402,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B = D )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).

thf(1344,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__decreasing_0) ).

thf(4942,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1344]) ).

thf(1361,axiom,
    tc_nat @ class_Ring__and__Field_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ozero__neq__one) ).

thf(4979,plain,
    tc_nat @ class_Ring__and__Field_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1361]) ).

thf(1453,axiom,
    ( c_Int_OMin
    = ( c_Int_OMin @ c_Int_OBit1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I43_J_1) ).

thf(5201,plain,
    ( c_Int_OMin
    = ( c_Int_OMin @ c_Int_OBit1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1453]) ).

thf(1609,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_4) ).

thf(5566,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1609]) ).

thf(224,axiom,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Opordered__ab__semigroup__add__imp__le) ).

thf(2327,plain,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(528,axiom,
    tc_Int_Oint @ class_OrderedGroup_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Omonoid__mult) ).

thf(3028,plain,
    tc_Int_Oint @ class_OrderedGroup_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(232,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__self1__is__0_0) ).

thf(2345,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(320,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__minus__divide__divide_0) ).

thf(2547,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(218,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mod__trivial_0) ).

thf(2313,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(841,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__nat_Odiff__0_0) ).

thf(3763,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(294,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( F @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right2_0) ).

thf(2492,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( F @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(989,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( B
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_square__eq__iff_0) ).

thf(4104,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( B
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(826,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( K @ ( D @ ( C @ ( J @ ( A @ ( I @ ( H @ ( B @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__less__power2_0) ).

thf(3727,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( K @ ( D @ ( C @ ( J @ ( A @ ( I @ ( H @ ( B @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(401,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__of__nonpos_0) ).

thf(2741,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(1255,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__sum__nat_3) ).

thf(4732,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1255]) ).

thf(183,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right__mono__neg_0) ).

thf(2228,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(793,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq__number__of_5) ).

thf(3647,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(134,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__self_0) ).

thf(2112,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(979,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__less__cancel1_1) ).

thf(4080,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(1796,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_4) ).

thf(5993,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1796]) ).

thf(916,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__nat__number__of_2) ).

thf(3931,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(42,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__one__right_0) ).

thf(1910,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(1233,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__left__cancel__less_0) ).

thf(4680,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1233]) ).

thf(856,axiom,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__Suc__0__nat_0) ).

thf(3799,plain,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(413,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I5_J_0) ).

thf(2769,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(708,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__mult__iff_4) ).

thf(3445,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(1078,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__one__odd__power_0) ).

thf(4316,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).

thf(1490,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__1_2) ).

thf(5285,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1490]) ).

thf(1447,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel2_2) ).

thf(5189,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1447]) ).

thf(1353,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( D @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower2__eq__square_0) ).

thf(4957,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( D @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1353]) ).

thf(1655,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower2__eq__square_0) ).

thf(5673,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1655]) ).

thf(1550,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__eq__number__of_1) ).

thf(5423,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1550]) ).

thf(1639,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__less__mono2_0) ).

thf(5635,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1639]) ).

thf(931,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ c_Int_OBit1 @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I4_J_1) ).

thf(3963,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ c_Int_OBit1 @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(1659,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mult__le__cancel1_0) ).

thf(5687,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1659]) ).

thf(933,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__power__eq__Suc__0__iff_1) ).

thf(3968,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(1503,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Oadd__left_0) ).

thf(5313,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1503]) ).

thf(189,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__positive__iff__positive_1) ).

thf(2245,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(297,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__add__num__eq__if__nat_0) ).

thf(2498,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(780,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__semidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__semidom) ).

thf(3615,plain,
    tc_nat @ class_Ring__and__Field_Oordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(870,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_9) ).

thf(3831,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(444,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__double__add__iff__zero__le__single__add_1) ).

thf(2840,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(135,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__less__1__iff_0) ).

thf(2115,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(1430,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__Numeral1_0) ).

thf(5145,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1430]) ).

thf(304,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of_2) ).

thf(2515,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(628,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__self_0) ).

thf(3254,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(129,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__le__zero__iff_1) ).

thf(2100,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(695,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__nonzero__iff__nonzero_0) ).

thf(3413,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(17,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__idempotent_0) ).

thf(1851,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(1410,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__le__cancel1_1) ).

thf(5098,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1410]) ).

thf(569,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__by__0_0) ).

thf(3122,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(246,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_3) ).

thf(2377,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(426,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right1_2) ).

thf(2801,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(1517,axiom,
    tc_Int_Oint @ ( c_Int_OPls @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I6_J_0) ).

thf(5345,plain,
    tc_Int_Oint @ ( c_Int_OPls @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1517]) ).

thf(1050,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__less__mono_0) ).

thf(4248,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(1630,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I2_J_0) ).

thf(5615,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1630]) ).

thf(1810,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Omul__d_0) ).

thf(6028,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1810]) ).

thf(237,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__minus_0) ).

thf(2359,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(1151,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__0__iff__number__of_1) ).

thf(4486,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).

thf(1582,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( L @ ( B @ ( A @ ( D @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Opower2__abs_0) ).

thf(5502,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( L @ ( B @ ( A @ ( D @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1582]) ).

thf(1708,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__eq__Suc__le_1) ).

thf(5798,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1708]) ).

thf(69,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right__le__one__le_0) ).

thf(1970,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(965,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Ocancel__semigroup__add) ).

thf(4048,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(1177,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__gt__zero__iff_1) ).

thf(4544,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1177]) ).

thf(1126,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Osemiring) ).

thf(4429,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).

thf(1213,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__Suc__eq__mod__add3_0) ).

thf(4636,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1213]) ).

thf(382,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus1__divide_0) ).

thf(2692,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(1168,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__diff__eq_0) ).

thf(4525,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).

thf(1303,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__left__cancel_0) ).

thf(4845,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1303]) ).

thf(1552,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__nat__equiv__def_1) ).

thf(5428,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1552]) ).

thf(1043,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__eq__power2_0) ).

thf(4233,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).

thf(29,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__self1_0) ).

thf(1878,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(395,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ring_Osub__add_0) ).

thf(2726,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(394,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_7) ).

thf(2724,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(1339,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( C @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__power__inverse_0) ).

thf(4930,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( C @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1339]) ).

thf(787,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__inverse__mult__distrib_0) ).

thf(3632,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(568,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__div__decomp_0) ).

thf(3119,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(1301,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__mult_0) ).

thf(4840,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1301]) ).

thf(1318,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__int__code_I16_J_1) ).

thf(4883,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1318]) ).

thf(1730,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__odd__power_1) ).

thf(5847,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1730]) ).

thf(794,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel1_2) ).

thf(3650,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(909,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I5_J_0) ).

thf(3916,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(921,axiom,
    ! [A: $i] :
      ( A
     != ( A @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_n__not__Suc__n_0) ).

thf(3941,plain,
    ! [A: $i] :
      ( A
     != ( A @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(1037,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__zero__eq_2) ).

thf(4219,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(199,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__divide_0) ).

thf(2268,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(706,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__le__imp__le_0) ).

thf(3440,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(1580,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left__mono_0) ).

thf(5497,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1580]) ).

thf(516,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__eq__0__iff_0) ).

thf(3001,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(1774,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__less__power_0) ).

thf(5951,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1774]) ).

thf(593,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__iff_3) ).

thf(3176,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(524,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__0_0) ).

thf(3018,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(1290,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__iff1_0) ).

thf(4817,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1290]) ).

thf(1324,axiom,
    ( ( tc_nat @ c_HOL_Oone__class_Oone )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Numeral1__eq1__nat_0) ).

thf(4898,plain,
    ( ( tc_nat @ c_HOL_Oone__class_Oone )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1324]) ).

thf(1231,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__nonneg__nonpos2_0) ).

thf(4676,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1231]) ).

thf(578,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right1_5) ).

thf(3143,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(783,axiom,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__numeral__1__eq__1_0) ).

thf(3622,plain,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(1014,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div_1) ).

thf(4167,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).

thf(263,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__nonzero__iff__nonzero_1) ).

thf(2419,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(1163,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__left__cancel__le_1) ).

thf(4513,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).

thf(701,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left2_5) ).

thf(3428,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(121,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__less__0__iff_0) ).

thf(2085,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(281,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div_0) ).

thf(2460,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(425,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__fieldgb_Odivide__inverse_0) ).

thf(2798,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(926,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_1) ).

thf(3954,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(966,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__Suc_0) ).

thf(4049,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(1770,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mult__less__cancel1_1) ).

thf(5941,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1770]) ).

thf(1221,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__0__iff__number__of_0) ).

thf(4653,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1221]) ).

thf(1723,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__le__zero__iff_2) ).

thf(5830,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1723]) ).

thf(577,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__mono_0) ).

thf(3141,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(1350,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__power__less__imp__less_0) ).

thf(4951,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1350]) ).

thf(652,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__nonpositive__iff__nonpositive_0) ).

thf(3311,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(658,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__by__1_0) ).

thf(3326,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(1438,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right__mono_0) ).

thf(5166,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1438]) ).

thf(1718,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right__less__imp__less_0) ).

thf(5818,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1718]) ).

thf(66,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__le__inverse__iff_1) ).

thf(1964,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(1062,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__Suc__Suc_0) ).

thf(4276,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(1191,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I4_J_1) ).

thf(4582,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).

thf(222,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__iff_0) ).

thf(2323,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(1363,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__nat__equiv__def_0) ).

thf(4983,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1363]) ).

thf(877,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( c_Suc @ hAPP ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__imp__le__base_0) ).

thf(3846,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( c_Suc @ hAPP ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(692,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_4) ).

thf(3407,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(1646,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__numeral__0__right_0) ).

thf(5651,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1646]) ).

thf(609,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__0__iff_4) ).

thf(3210,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(1633,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__one__right_0) ).

thf(5620,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1633]) ).

thf(772,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_trans__less__add2_0) ).

thf(3596,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(1123,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__leE_1) ).

thf(4421,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).

thf(520,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__le__not__le_2) ).

thf(3010,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(119,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide_Oadd_0) ).

thf(2080,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(1475,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__imp__less__div__pos_0) ).

thf(5253,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1475]) ).

thf(169,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right2_2) ).

thf(2196,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(187,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__ge__zero_0) ).

thf(2240,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(11,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
       != B )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__equal__zero_0) ).

thf(1842,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
       != B )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(1600,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__mult__less_0) ).

thf(5547,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1600]) ).

thf(1458,axiom,
    ! [A: $i] :
      ( ( A @ c_Int_OBit1 )
     != c_Int_OPls ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I46_J_0) ).

thf(5213,plain,
    ! [A: $i] :
      ( ( A @ c_Int_OBit1 )
     != c_Int_OPls ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1458]) ).

thf(530,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__eq__iff_0) ).

thf(3032,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(53,axiom,
    tc_Int_Oint @ class_OrderedGroup_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Oab__semigroup__mult) ).

thf(1937,plain,
    tc_Int_Oint @ class_OrderedGroup_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(791,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__commutes_0) ).

thf(3643,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(1311,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__minus1_0) ).

thf(4867,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1311]) ).

thf(1754,axiom,
    tc_nat @ class_OrderedGroup_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Oab__semigroup__mult) ).

thf(5905,plain,
    tc_nat @ class_OrderedGroup_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1754]) ).

thf(33,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right1_5) ).

thf(1889,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(755,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( A @ ( K @ ( C @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( G = B )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__gt__zero__iff_1) ).

thf(3555,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( A @ ( K @ ( C @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( G = B )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(1138,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__nat__equiv__def2_1) ).

thf(4457,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1138]) ).

thf(506,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__divide__divide_0) ).

thf(2978,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(363,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__mod_5) ).

thf(2649,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(645,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Oabs__if ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__if_1) ).

thf(3294,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Oabs__if ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(1535,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__nat__eq_0) ).

thf(5387,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1535]) ).

thf(1257,axiom,
    ! [A: $i] :
      ( ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oone__class_Oone @ c_Power_Opower_Opower ) ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__power__power_0) ).

thf(4736,plain,
    ! [A: $i] :
      ( ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oone__class_Oone @ c_Power_Opower_Opower ) ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1257]) ).

thf(418,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__div__pos_0) ).

thf(2781,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(1170,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__RealVector_Oreal__normed__algebra) ).

thf(4530,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).

thf(918,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__comm__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__strict__left__mono__comm_0) ).

thf(3935,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__comm__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(1485,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__div__cancel__disj_0) ).

thf(5273,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1485]) ).

thf(139,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right2_3) ).

thf(2125,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(1108,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__0_0) ).

thf(4384,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).

thf(919,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__asym_0) ).

thf(3937,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(58,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__self__if_0) ).

thf(1947,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(76,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__le__trans_0) ).

thf(1984,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(1801,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__odd__eq_0) ).

thf(6005,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1801]) ).

thf(1528,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq__number__of_1) ).

thf(5369,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1528]) ).

thf(145,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__minus__iff_1) ).

thf(2140,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(1472,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I11_J_0) ).

thf(5245,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1472]) ).

thf(1285,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__diff__less__iff_1) ).

thf(4804,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1285]) ).

thf(622,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__add__le__zero__iff__single__add__le__zero_0) ).

thf(3242,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(815,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I3_J_0) ).

thf(3703,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(206,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__minus__self__iff_1) ).

thf(2285,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(1614,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_division__ring__inverse__diff_0) ).

thf(5576,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1614]) ).

thf(501,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__le__divisor_0) ).

thf(2968,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(1397,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Divides_Omod__div__equality_H_0) ).

thf(5063,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1397]) ).

thf(314,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__iff_4) ).

thf(2536,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(749,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power2_0) ).

thf(3541,plain,
    ! [A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(292,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__one__less__zero_0) ).

thf(2487,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(542,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__add__less__zero__iff__single__less__zero_0) ).

thf(3059,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(250,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__le__1__iff_1) ).

thf(2387,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(151,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_frac__eq__eq_0) ).

thf(2153,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(472,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__0__le__iff__le_0) ).

thf(2900,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(1641,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = F )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__add__iff1_0) ).

thf(5639,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = F )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1641]) ).

thf(1426,axiom,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__num__eq__if__nat_1) ).

thf(5134,plain,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1426]) ).

thf(845,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I37_J_0) ).

thf(3770,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(71,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__diff__triangle__ineq_0) ).

thf(1974,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(51,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_xt1_I8_J_0) ).

thf(1932,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(1048,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__even__eq_0) ).

thf(4244,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).

thf(334,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__self_0) ).

thf(2581,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(283,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__le__iff__le_0) ).

thf(2465,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(1341,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__right__cancel_0) ).

thf(4934,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1341]) ).

thf(1175,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
      = ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Fact_Ofact__Suc_0) ).

thf(4540,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
      = ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).

thf(85,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__le__0__iff__le_1) ).

thf(2006,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(1042,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__1__right_0) ).

thf(4231,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(432,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__inverse_0) ).

thf(2813,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(1091,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_no__zero__divirors__neq0_0) ).

thf(4347,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).

thf(693,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_6) ).

thf(3409,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(455,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__divide__distrib_0) ).

thf(2862,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(106,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_8) ).

thf(2052,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(1326,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__distrib_0) ).

thf(4902,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1326]) ).

thf(661,axiom,
    ! [A: $i] :
      ( ( ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_number__of__Min_0) ).

thf(3333,plain,
    ! [A: $i] :
      ( ( ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(1351,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ c_Int_OBit1 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I9_J_0) ).

thf(4953,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ c_Int_OBit1 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1351]) ).

thf(564,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__of__nonneg_0) ).

thf(3109,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(1237,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat_Osimps_I2_J_0) ).

thf(4689,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1237]) ).

thf(911,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__minus_0) ).

thf(3920,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(1120,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__lemma_0) ).

thf(4416,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).

thf(673,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__iff_4) ).

thf(3364,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(605,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_division__ring__inverse__add_0) ).

thf(3201,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(1039,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I16_J_0) ).

thf(4223,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(722,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__le__iff_2) ).

thf(3476,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(1173,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq_4) ).

thf(4535,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).

thf(615,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__nonneg__pos_0) ).

thf(3225,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(1276,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_iszero__number__of__Bit0_0) ).

thf(4782,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1276]) ).

thf(1546,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nat__number__of_0) ).

thf(5415,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1546]) ).

thf(1803,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__div__cancel_0) ).

thf(6011,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1803]) ).

thf(256,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right2_4) ).

thf(2402,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( D @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(1328,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__nonneg__nonpos_0) ).

thf(4906,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1328]) ).

thf(180,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__minus__iff_0) ).

thf(2221,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(36,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq_2) ).

thf(1895,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(310,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__nonneg__neg_0) ).

thf(2529,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(1440,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__increasing__iff_0) ).

thf(5170,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1440]) ).

thf(580,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__le__less_1) ).

thf(3147,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(712,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__strict__increasing_0) ).

thf(3454,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(357,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__number__of_1) ).

thf(2634,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(19,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div_5) ).

thf(1857,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(1605,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__one__le__power_0) ).

thf(5557,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1605]) ).

thf(1021,axiom,
    ! [A: $i] :
      ( ( A = c_Int_OMin )
      | ( ( A @ c_Int_OBit1 )
       != c_Int_OMin ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I47_J_0) ).

thf(4184,plain,
    ! [A: $i] :
      ( ( A = c_Int_OMin )
      | ( ( A @ c_Int_OBit1 )
       != c_Int_OMin ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).

thf(1417,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_half_0) ).

thf(5112,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1417]) ).

thf(404,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__equal__0__iff__equal_0) ).

thf(2749,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(1502,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__Suc2_0) ).

thf(5310,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1502]) ).

thf(1433,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__power__eq__Suc__0__iff_0) ).

thf(5153,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1433]) ).

thf(1093,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__left__cancel__le_0) ).

thf(4352,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).

thf(172,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( D @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( B @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( D @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Rational_Oordered__idom_Osgn__greater_0) ).

thf(2203,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( D @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( B @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( D @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(1670,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__eq_1) ).

thf(5711,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1670]) ).

thf(758,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left_Ominus_0) ).

thf(3563,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(409,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Ocomm__semiring) ).

thf(2761,plain,
    tc_Int_Oint @ class_Ring__and__Field_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(1025,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__imp__less__left_0) ).

thf(4195,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).

thf(784,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__diff__add__eq1_0) ).

thf(3624,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(1209,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_9) ).

thf(4625,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1209]) ).

thf(1407,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__strict__increasing__iff_1) ).

thf(5090,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1407]) ).

thf(768,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mod__equality_H_0) ).

thf(3587,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(596,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_6) ).

thf(3183,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(1356,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left__less__imp__less_0) ).

thf(4966,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1356]) ).

thf(1269,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__mult__neg__le_0) ).

thf(4765,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1269]) ).

thf(592,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
        = C )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__unique_0) ).

thf(3173,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
        = C )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(1147,axiom,
    ! [A: $i] :
      ( ( A = c_Int_OPls )
      | ( ( A @ c_Int_OBit0 )
       != c_Int_OPls ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I44_J_0) ).

thf(4476,plain,
    ! [A: $i] :
      ( ( A = c_Int_OPls )
      | ( ( A @ c_Int_OBit0 )
       != c_Int_OPls ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).

thf(650,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__leE_0) ).

thf(3306,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(13,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__le__D1_0) ).

thf(1846,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(1555,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I8_J_0) ).

thf(5434,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1555]) ).

thf(1781,axiom,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__ge__Suc__0__nat_0) ).

thf(5964,plain,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1781]) ).

thf(498,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_4) ).

thf(2963,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(1560,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__iszero__Numeral1_0) ).

thf(5447,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1560]) ).

thf(361,axiom,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__1__nat_0) ).

thf(2645,plain,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(1586,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__m1__even_0) ).

thf(5513,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1586]) ).

thf(1451,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__increasing_0) ).

thf(5198,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1451]) ).

thf(875,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__sum__squares__lt__zero_0) ).

thf(3841,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(350,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__number__of__Min_0) ).

thf(2617,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(1423,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__less_0) ).

thf(5126,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1423]) ).

thf(1727,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__le__imp__less_0) ).

thf(5840,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1727]) ).

thf(369,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__frac__frac_0) ).

thf(2662,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(948,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__strict__left__mono__neg_0) ).

thf(4004,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(229,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__0_0) ).

thf(2338,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(804,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__iff1_0) ).

thf(3675,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(1074,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mult__less__cancel1_0) ).

thf(4308,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(477,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( F @ ( A @ ( L @ ( K @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( G = B )
      | ( H @ ( G @ ( F @ hAPP ) @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__less__power__abs__iff_1) ).

thf(2909,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( F @ ( A @ ( L @ ( K @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( G = B )
      | ( H @ ( G @ ( F @ hAPP ) @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(1179,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__nat__number__of_1) ).

thf(4550,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1179]) ).

thf(147,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq_3) ).

thf(2144,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(83,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I6_J_0) ).

thf(2001,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(1652,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( B @ ( A @ ( K @ ( D @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( C @ ( D @ hAPP ) @ ( C @ ( B @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__sgn__abs_0) ).

thf(5666,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( B @ ( A @ ( K @ ( D @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( C @ ( D @ hAPP ) @ ( C @ ( B @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1652]) ).

thf(1235,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I10_J_0) ).

thf(4685,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1235]) ).

thf(252,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power_0) ).

thf(2391,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(535,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left1_4) ).

thf(3042,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(729,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__distrib__number__of_0) ).

thf(3491,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(1466,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__less__cancel__left_1) ).

thf(5230,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1466]) ).

thf(1111,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I10_J_0) ).

thf(4393,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).

thf(469,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_2) ).

thf(2894,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(665,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__minus__cancel_0) ).

thf(3343,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(1335,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__if_0) ).

thf(4921,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1335]) ).

thf(40,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__less__1__iff_1) ).

thf(1906,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(1657,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__self__is__m_0) ).

thf(5679,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1657]) ).

thf(396,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of1_3) ).

thf(2729,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(1334,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__diff__split__asm_1) ).

thf(4919,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1334]) ).

thf(572,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_frac__eq__eq_1) ).

thf(3130,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( E
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(1676,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__pred_H_0) ).

thf(5722,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1676]) ).

thf(999,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__Suc_0) ).

thf(4131,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(1714,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_termination__basic__simps_I5_J_0) ).

thf(5808,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1714]) ).

thf(63,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__1__iff_1) ).

thf(1959,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(324,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left2_3) ).

thf(2556,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(1094,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__int__code_I15_J_0) ).

thf(4354,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).

thf(192,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right1_2) ).

thf(2253,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(98,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left__mono__neg_0) ).

thf(2033,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(149,axiom,
    ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I7_J_0) ).

thf(2149,plain,
    ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(1196,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__sum__nat_2) ).

thf(4594,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1196]) ).

thf(1784,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Omul__1_0) ).

thf(5969,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1784]) ).

thf(629,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__eq__divide_0) ).

thf(3257,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(1562,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I11_J_0) ).

thf(5451,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1562]) ).

thf(1314,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_lemma__even__div2_0) ).

thf(4874,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1314]) ).

thf(1584,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__diff__conv2_0) ).

thf(5508,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1584]) ).

thf(330,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_xt1_I11_J_0) ).

thf(2570,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(490,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( D @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__minus__cong_0) ).

thf(2941,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( D @ c_HOL_Ouminus__class_Ouminus ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(1184,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__eq__1__iff_1) ).

thf(4562,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).

thf(631,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Omult__zero) ).

thf(3262,plain,
    tc_Int_Oint @ class_Ring__and__Field_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(1627,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__0__Suc_0) ).

thf(5608,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1627]) ).

thf(343,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__less__inverse__iff_0) ).

thf(2602,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(96,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__eq__mult_2) ).

thf(2028,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(398,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__less__imp__less_0) ).

thf(2734,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(1765,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( E @ ( C @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I29_J_0) ).

thf(5928,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( E @ ( C @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1765]) ).

thf(1376,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__eqI_0) ).

thf(5016,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1376]) ).

thf(1745,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I23_J_0) ).

thf(5882,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1745]) ).

thf(1409,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I12_J_0) ).

thf(5095,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1409]) ).

thf(1059,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult2__eq_0) ).

thf(4269,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(1644,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__le__less__mono_0) ).

thf(5647,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1644]) ).

thf(953,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq__number__of_1) ).

thf(4017,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(1672,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__strict__left__mono__neg_0) ).

thf(5715,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1672]) ).

thf(1152,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__commute_0) ).

thf(4489,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).

thf(1819,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__imp__less__exp_0) ).

thf(6052,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1819]) ).

thf(688,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__abs__iff_1) ).

thf(3396,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(245,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Osemiring) ).

thf(2376,plain,
    tc_Int_Oint @ class_Ring__and__Field_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(548,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__le__not__le_1) ).

thf(3070,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(502,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_positive__imp__inverse__positive_0) ).

thf(2970,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(293,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = C )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of_0) ).

thf(2489,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = C )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(1779,axiom,
    tc_Complex_Ocomplex @ class_Int_Onumber,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Int_Onumber) ).

thf(5962,plain,
    tc_Complex_Ocomplex @ class_Int_Onumber,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1779]) ).

thf(1142,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__less__Suc__eq_1) ).

thf(4465,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).

thf(714,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__neg__pos_0) ).

thf(3458,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(1083,axiom,
    tc_nat @ class_Int_Onumber,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Int_Onumber) ).

thf(4328,plain,
    tc_nat @ class_Int_Onumber,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).

thf(383,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__eq__0__iff_1) ).

thf(2695,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(1664,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_6) ).

thf(5698,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1664]) ).

thf(1004,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__0__less__mult__iff_1) ).

thf(4144,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(676,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div_H_6) ).

thf(3372,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(946,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__number__of__eq__not__less_0) ).

thf(4000,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(683,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_7) ).

thf(3387,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(1224,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_8) ).

thf(4662,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1224]) ).

thf(1564,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div2__Suc__Suc_0) ).

thf(5456,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1564]) ).

thf(913,axiom,
    ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I2_J_0) ).

thf(3925,plain,
    ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(453,axiom,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Opordered__ab__semigroup__add) ).

thf(2859,plain,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(489,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Oadd__a_0) ).

thf(2938,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(1571,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__m1__odd_0) ).

thf(5474,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1571]) ).

thf(182,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__add__cong_0) ).

thf(2225,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(734,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__eq__eq_0) ).

thf(3504,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(1065,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I21_J_0) ).

thf(4283,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(1576,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I8_J_1) ).

thf(5488,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1576]) ).

thf(1154,axiom,
    tc_nat @ class_Ring__and__Field_Omult__mono,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Omult__mono) ).

thf(4494,plain,
    tc_nat @ class_Ring__and__Field_Omult__mono,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).

thf(861,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__iff_0) ).

thf(3811,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(1538,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__inject__exp_0) ).

thf(5395,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1538]) ).

thf(621,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__one_0) ).

thf(3240,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(421,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( B @ ( I @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( L @ ( A @ ( K @ ( D @ ( J @ ( I @ ( H @ ( G @ ( F @ ( C @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( B @ ( A @ hAPP ) )
       != ( D @ ( C @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__1__neg_0) ).

thf(2789,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( B @ ( I @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( L @ ( A @ ( K @ ( D @ ( J @ ( I @ ( H @ ( G @ ( F @ ( C @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( B @ ( A @ hAPP ) )
       != ( D @ ( C @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(740,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I17_J_0) ).

thf(3519,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(1729,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__0__iff_0) ).

thf(5844,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1729]) ).

thf(778,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__diff__inverse_0) ).

thf(3609,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(448,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__le__zero__iff_0) ).

thf(2849,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(308,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_1) ).

thf(2525,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(1398,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( M @ ( F @ ( E @ ( L @ ( K @ ( C @ ( D @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__le__zero__iff_1) ).

thf(5065,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( M @ ( F @ ( E @ ( L @ ( K @ ( C @ ( D @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1398]) ).

thf(165,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__add__less__zero__iff__single__less__zero_1) ).

thf(2188,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(507,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__right__mono_0) ).

thf(2981,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(253,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of1_4) ).

thf(2393,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(555,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__mult__iff_0) ).

thf(3086,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(830,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Omul__1_0) ).

thf(3736,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(839,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__mult_0) ).

thf(3758,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(1636,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( K @ ( A @ ( C @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__ge__zero_0) ).

thf(5627,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( K @ ( A @ ( C @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1636]) ).

thf(633,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__self__if_1) ).

thf(3266,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(374,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( E @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right2_1) ).

thf(2674,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( E @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(329,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right1_1) ).

thf(2568,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(1629,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__mult__left_0) ).

thf(5612,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1629]) ).

thf(959,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__diff__eq__diff__pred_0) ).

thf(4032,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(429,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__0__iff_0) ).

thf(2806,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(511,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oring) ).

thf(2991,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(776,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_1) ).

thf(3604,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(1106,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__even__mod__4__iff_1) ).

thf(4381,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).

thf(1484,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I3_J_0) ).

thf(5271,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1484]) ).

thf(691,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__commutes_0) ).

thf(3404,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(887,axiom,
    ( c_Int_OPls
    = ( c_Int_OPls @ c_Int_OBit0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I38_J_1) ).

thf(3868,plain,
    ( c_Int_OPls
    = ( c_Int_OPls @ c_Int_OBit0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(1266,axiom,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
      = ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__eq__plus1_0) ).

thf(4758,plain,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
      = ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1266]) ).

thf(1271,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__gt__zero__iff_0) ).

thf(4769,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1271]) ).

thf(1160,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__le_1) ).

thf(4505,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).

thf(774,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_5) ).

thf(3599,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(1782,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_7) ).

thf(5965,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1782]) ).

thf(526,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__trans_0) ).

thf(3023,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(846,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I19_J_0) ).

thf(3773,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(836,axiom,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__add__less2_0) ).

thf(3751,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(1525,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mono_0) ).

thf(5363,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1525]) ).

thf(664,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__neg__iff__add__eq__0_0) ).

thf(3340,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(1431,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__add__num__eq__if__nat_1) ).

thf(5148,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1431]) ).

thf(1061,axiom,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__power2_0) ).

thf(4273,plain,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(648,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__leI_0) ).

thf(3302,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(614,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_3) ).

thf(3223,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(154,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_semiring__div__class_Omod__div__equality_H_0) ).

thf(2161,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(1282,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__is__add_3) ).

thf(4796,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1282]) ).

thf(468,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq_3) ).

thf(2891,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(1300,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( F @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I20_J_0) ).

thf(4837,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( F @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1300]) ).

thf(551,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( A @ ( K @ ( D @ ( J @ ( E @ ( I @ ( H @ ( F @ ( C @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( ( B @ ( A @ hAPP ) )
        = ( D @ ( C @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__neg_0) ).

thf(3077,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( A @ ( K @ ( D @ ( J @ ( E @ ( I @ ( H @ ( F @ ( C @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( ( B @ ( A @ hAPP ) )
        = ( D @ ( C @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(1218,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__lessD1_0) ).

thf(4645,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1218]) ).

thf(792,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Osemiring__0) ).

thf(3646,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(992,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_expand__Suc_0) ).

thf(4112,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(1735,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__zero__less__power__iff_0) ).

thf(5859,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1735]) ).

thf(812,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__mono_H_0) ).

thf(3695,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(987,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Ominus__left_0) ).

thf(4099,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(660,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__divide__mult__cancel__left_0) ).

thf(3330,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(400,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__antisym_0) ).

thf(2738,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(1006,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBC ) ) ) ) ) )
      = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ATP__Linkup_OCOMBC__def_0) ).

thf(4149,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBC ) ) ) ) ) )
      = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(608,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__ge__zero_0) ).

thf(3208,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(1489,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( D @ ( C @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_lemma__realpow__diff_0) ).

thf(5282,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( D @ ( C @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1489]) ).

thf(1044,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Odiff__right_0) ).

thf(4236,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(799,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__mono2_0) ).

thf(3663,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(488,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I4_J_0) ).

thf(2935,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(1293,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right_Oadd_0) ).

thf(4822,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1293]) ).

thf(272,axiom,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Opordered__ab__group__add) ).

thf(2442,plain,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(408,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__iff_0) ).

thf(2759,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(850,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__diff__inverse_0) ).

thf(3783,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(727,axiom,
    tc_Int_Oint @ class_Orderings_Oorder,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Orderings_Oorder) ).

thf(3487,plain,
    tc_Int_Oint @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(108,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of_1) ).

thf(2055,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(387,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__minus__self__iff_0) ).

thf(2706,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(1561,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__iff__diff__less__0_1) ).

thf(5449,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1561]) ).

thf(156,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_2) ).

thf(2166,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(782,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__odd__eq_0) ).

thf(3619,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(764,axiom,
    tc_nat @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add) ).

thf(3579,plain,
    tc_nat @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(279,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oordered__semiring) ).

thf(2457,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(1306,axiom,
    ! [A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__0__left_0) ).

thf(4854,plain,
    ! [A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1306]) ).

thf(265,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_5) ).

thf(2425,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(696,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Lim_Ominus__diff__minus_0) ).

thf(3416,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(1798,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__diff__conv2_1) ).

thf(5997,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1798]) ).

thf(649,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left2_1) ).

thf(3304,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(1702,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I1_J_0) ).

thf(5784,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1702]) ).

thf(287,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__negative__iff__negative_1) ).

thf(2474,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(1320,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Ominus__right_0) ).

thf(4887,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1320]) ).

thf(1554,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__diff__iff_0) ).

thf(5432,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1554]) ).

thf(331,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__divide__right_0) ).

thf(2573,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(220,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_9) ).

thf(2319,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(302,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__number__of_0) ).

thf(2510,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(1249,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__antisym__conv3_0) ).

thf(4716,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1249]) ).

thf(1808,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( H @ ( G @ ( D @ ( F @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__ring__1_Opower2__minus_0) ).

thf(6024,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( H @ ( G @ ( D @ ( F @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1808]) ).

thf(393,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__eq__1__iff_0) ).

thf(2721,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(315,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__semidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oordered__semidom) ).

thf(2538,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(697,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_0) ).

thf(3419,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(828,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__0__less__mult__iff_0) ).

thf(3732,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(212,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__iff_1) ).

thf(2299,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(1193,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left_Ozero_0) ).

thf(4586,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).

thf(818,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__diff__inverse2_0) ).

thf(3710,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(1785,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I2_J_1) ).

thf(5972,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1785]) ).

thf(990,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__less__Suc__eq_0) ).

thf(4107,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(112,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__minus__one_0) ).

thf(2065,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(561,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__nonnegative__iff__nonnegative_0) ).

thf(3101,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(306,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__pos__pos_0) ).

thf(2521,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(1279,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( L @ ( D @ ( C @ ( K @ ( J @ ( A @ ( I @ ( B @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__le__power2_0) ).

thf(4790,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( L @ ( D @ ( C @ ( K @ ( J @ ( A @ ( I @ ( B @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( F @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1279]) ).

thf(1087,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I10_J_0) ).

thf(4337,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).

thf(376,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__one_0) ).

thf(2679,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(1654,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_4) ).

thf(5671,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1654]) ).

thf(1157,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Opwr__Suc_0) ).

thf(4498,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1157]) ).

thf(1101,axiom,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gr0I_0) ).

thf(4369,plain,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).

thf(150,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( B @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left2_0) ).

thf(2151,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( B @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(360,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__self2_0) ).

thf(2642,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(584,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__positive__iff__positive_0) ).

thf(3155,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(863,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1__no__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_field__power__not__zero_0) ).

thf(3815,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(177,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_0) ).

thf(2214,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(353,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__by__1_0) ).

thf(2624,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(1413,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I36_J_0) ).

thf(5103,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1413]) ).

thf(1589,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left_Oadd_0) ).

thf(5520,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1589]) ).

thf(753,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__int__code_I16_J_0) ).

thf(3551,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(264,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_1) ).

thf(2422,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(533,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_3) ).

thf(3038,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(358,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_6) ).

thf(2637,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(928,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__0__iff_2) ).

thf(3957,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(415,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nonneg__pos_0) ).

thf(2774,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(642,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__ring__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oordered__ring__strict) ).

thf(3287,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oordered__ring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(1296,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__less__linear_0) ).

thf(4828,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1296]) ).

thf(1499,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq_1) ).

thf(5303,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1499]) ).

thf(1722,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__minus_0) ).

thf(5827,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1722]) ).

thf(127,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_Int_Oiszero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__iszero__1_0) ).

thf(2097,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_Int_Oiszero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(1733,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__diff_0) ).

thf(5853,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1733]) ).

thf(635,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add__abs )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__if__lattice_1) ).

thf(3270,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add__abs )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(1199,axiom,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Fact_Ofact__0__nat_0) ).

thf(4601,plain,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).

thf(563,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__minus__eq_0) ).

thf(3106,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(718,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_9) ).

thf(3468,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(1393,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__even__abs_0) ).

thf(5054,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1393]) ).

thf(372,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_4) ).

thf(2669,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(1461,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( B @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__div__eq__add3__div_0) ).

thf(5222,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( B @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1461]) ).

thf(1034,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__sum__nat_0) ).

thf(4213,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).

thf(1205,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__diff__conv_0) ).

thf(4616,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1205]) ).

thf(603,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__neg__nonpos_0) ).

thf(3197,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(1518,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__diff__le_0) ).

thf(5346,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1518]) ).

thf(976,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_iszero__number__of__Pls_0) ).

thf(4073,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(1593,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__less__mono1_0) ).

thf(5531,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1593]) ).

thf(451,axiom,
    tc_Int_Oint @ class_Divides_Oring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Divides_Oring__div) ).

thf(2856,plain,
    tc_Int_Oint @ class_Divides_Oring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(1018,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power2_0) ).

thf(4176,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).

thf(1660,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__Suc__eq__0__disj_3) ).

thf(5689,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1660]) ).

thf(1071,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq_2) ).

thf(4300,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(1103,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__add__iff1_0) ).

thf(4374,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).

thf(546,axiom,
    tc_Int_Oint @ class_OrderedGroup_Ogroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Ogroup__add) ).

thf(3066,plain,
    tc_Int_Oint @ class_OrderedGroup_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(470,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__iff_5) ).

thf(2896,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(713,axiom,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_Parity_Oeven__odd__class_Oeven ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__1__nat_0) ).

thf(3456,plain,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_Parity_Oeven__odd__class_Oeven ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(834,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__strict__mono_0) ).

thf(3747,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(356,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__less__inverse__iff_2) ).

thf(2632,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(573,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left1_4) ).

thf(3133,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(1404,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__1__right_0) ).

thf(5082,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1404]) ).

thf(1442,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__num__iff_0) ).

thf(5175,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1442]) ).

thf(407,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__minus__self__iff_0) ).

thf(2757,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(1020,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ ( c_Suc @ hAPP ) )
       != ( B @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__inject_0) ).

thf(4181,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ ( c_Suc @ hAPP ) )
       != ( B @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).

thf(518,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Ono__zero__divisors) ).

thf(3006,plain,
    tc_Int_Oint @ class_Ring__and__Field_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(1568,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__mult__commute_0) ).

thf(5466,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1568]) ).

thf(884,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = D ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__one__right_0) ).

thf(3861,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(333,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__eq__zero__iff_0) ).

thf(2578,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(1603,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__mult1_0) ).

thf(5552,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1603]) ).

thf(437,axiom,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__num__eq__if__nat_0) ).

thf(2824,plain,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Fact_Ofact__class_Ofact ) )
    = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(1029,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq__number__of_4) ).

thf(4203,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).

thf(1418,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__one__odd__power_0) ).

thf(5114,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( B @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1418]) ).

thf(163,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__div__trivial_0) ).

thf(2182,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(1527,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__zero__eq_0) ).

thf(5367,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1527]) ).

thf(1127,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_0) ).

thf(4430,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).

thf(1617,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A
        = ( B @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__SucE_0) ).

thf(5584,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A
        = ( B @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1617]) ).

thf(1761,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__2_0) ).

thf(5920,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1761]) ).

thf(1122,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add1_0) ).

thf(4420,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).

thf(1133,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__2__gt__zero_0) ).

thf(4445,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).

thf(345,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__mod_0) ).

thf(2605,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(1679,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gr0__conv__Suc_1) ).

thf(5730,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1679]) ).

thf(801,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_termination__basic__simps_I2_J_0) ).

thf(3668,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(1141,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_combine__common__factor_0) ).

thf(4462,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1141]) ).

thf(1330,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__imp__le__exp_0) ).

thf(4911,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1330]) ).

thf(195,axiom,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__ge__one__nat_0) ).

thf(2261,plain,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(1,negated_conjecture,
    tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Oord__class_Oless ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_conjecture_0) ).

thf(1821,plain,
    tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).

thf(1057,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__is__0_2) ).

thf(4264,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(761,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__strict__left__mono_0) ).

thf(3572,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(1136,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide_Odiff_0) ).

thf(4452,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).

thf(1623,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__right__cancel_0) ).

thf(5599,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1623]) ).

thf(531,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__iff_0) ).

thf(3034,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(9,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Oabs__if,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Oabs__if) ).

thf(1838,plain,
    tc_Int_Oint @ class_Ring__and__Field_Oabs__if,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(1648,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__zero__left_0) ).

thf(5656,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1648]) ).

thf(1505,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( B @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I8_J_0) ).

thf(5318,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( B @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1505]) ).

thf(460,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_5) ).

thf(2874,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(1746,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I6_J_0) ).

thf(5885,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1746]) ).

thf(725,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__le__zero__iff_1) ).

thf(3483,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(766,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__square_0) ).

thf(3583,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(1816,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__pos_0) ).

thf(6044,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1816]) ).

thf(362,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_Int_Oiszero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_iszero__0_0) ).

thf(2647,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_Int_Oiszero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(208,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__even__power_0) ).

thf(2289,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(1444,axiom,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) )
        = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mult__two__diff__one_0) ).

thf(5180,plain,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) )
        = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1444]) ).

thf(203,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( B @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( D @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Rational_Oordered__idom_Osgn__less_1) ).

thf(2278,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( B @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( D @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(1372,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power_Opower_Opower__0_0) ).

thf(5007,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1372]) ).

thf(504,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__less__1__iff_2) ).

thf(2974,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(1544,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__leI_0) ).

thf(5410,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1544]) ).

thf(445,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__less__eq__nonneg_0) ).

thf(2842,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Oordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(1726,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__le__imp__le_0) ).

thf(5838,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1726]) ).

thf(60,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_1) ).

thf(1952,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(1700,axiom,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
      = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc3__eq__add__3_0) ).

thf(5780,plain,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
      = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1700]) ).

thf(1181,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__1_4) ).

thf(4554,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).

thf(724,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_group__add__class_Odiff__0_0) ).

thf(3480,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(1684,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__neg__neg_0) ).

thf(5742,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1684]) ).

thf(1766,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( A @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( B @ ( A @ hAPP ) @ ( A @ hAPP ) )
        = ( B @ ( A @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__sgn_0) ).

thf(5931,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( A @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( B @ ( A @ hAPP ) @ ( A @ hAPP ) )
        = ( B @ ( A @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1766]) ).

thf(1574,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__diff__cancel_0) ).

thf(5482,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1574]) ).

thf(1159,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__is__add_0) ).

thf(4502,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).

thf(1587,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_0) ).

thf(5516,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1587]) ).

thf(486,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__less__0__iff__less_1) ).

thf(2931,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(523,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_8) ).

thf(3016,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(1731,axiom,
    ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__1__add__1_0) ).

thf(5849,plain,
    ( ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1731]) ).

thf(1243,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_9) ).

thf(4703,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1243]) ).

thf(500,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le0_0) ).

thf(2967,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(214,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__left__mono_0) ).

thf(2303,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(905,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div__lemma_1) ).

thf(3908,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(457,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__0__right_0) ).

thf(2867,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(748,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_6) ).

thf(3539,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(116,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__0__less__iff__less_1) ).

thf(2075,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(1340,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__le__self_0) ).

thf(4933,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1340]) ).

thf(1394,axiom,
    ( ( c_Int_OMin @ c_Int_OBit1 )
    = c_Int_OMin ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Bit1__Min_0) ).

thf(5057,plain,
    ( ( c_Int_OMin @ c_Int_OBit1 )
    = c_Int_OMin ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1394]) ).

thf(271,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_left__inverse_0) ).

thf(2439,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(785,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__eq__add__iff2_0) ).

thf(3627,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(1529,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__if_1) ).

thf(5372,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1529]) ).

thf(158,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__iff_0) ).

thf(2171,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(1459,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__times__mod__eq_0) ).

thf(5216,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1459]) ).

thf(873,axiom,
    ~ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_False_0) ).

thf(3838,plain,
    ~ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(411,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_4) ).

thf(2765,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(321,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( L @ ( G @ hAPP ) @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__less__power__abs__iff_2) ).

thf(2550,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( L @ ( G @ hAPP ) @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(290,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ c_Int_Oiszero ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_iszero__def_0) ).

thf(2481,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ c_Int_Oiszero ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(809,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__nat__div__two__times__two__plus__one_0) ).

thf(3687,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(1473,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__diff__assoc2_0) ).

thf(5248,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1473]) ).

thf(859,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I14_J_1) ).

thf(3806,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(586,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__le__self__iff_0) ).

thf(3159,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(242,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of1_2) ).

thf(2369,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(1098,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__int__code_I15_J_1) ).

thf(4362,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).

thf(339,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__equation__iff_1) ).

thf(2592,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(405,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__eq__iff__eq_0) ).

thf(2752,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(816,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__Suc__right_0) ).

thf(3706,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(1677,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Onoteq__reduce_1) ).

thf(5725,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1677]) ).

thf(1809,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Omonoid__add) ).

thf(6027,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1809]) ).

thf(594,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_7) ).

thf(3178,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(1541,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add__diff_0) ).

thf(5403,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1541]) ).

thf(515,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__strict__right__mono__neg_0) ).

thf(2999,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(1662,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__diff__less__iff_2) ).

thf(5693,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1662]) ).

thf(667,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__add__self1_0) ).

thf(3348,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(152,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_equation__minus__iff_1) ).

thf(2156,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(1579,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_n__less__n__mult__m_0) ).

thf(5495,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1579]) ).

thf(974,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_lemma__even__mod__4__div__2_0) ).

thf(4067,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(174,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_0) ).

thf(2207,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(1508,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__less__divisor_0) ).

thf(5326,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1508]) ).

thf(1321,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__power__le__0__imp__0_0) ).

thf(4890,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1321]) ).

thf(1149,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__0__iff_1) ).

thf(4481,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).

thf(1010,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__pos_0) ).

thf(4158,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).

thf(300,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_0) ).

thf(2506,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(680,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__Suc2_0) ).

thf(3380,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(981,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__nat__eq__number__of_0) ).

thf(4084,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(1778,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_number__of__Bit0_0) ).

thf(5959,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1778]) ).

thf(601,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__eq__neg_0) ).

thf(3193,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(132,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_0) ).

thf(2107,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(587,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of1_1) ).

thf(3161,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(1079,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_2) ).

thf(4319,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).

thf(1316,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__mult__distrib_0) ).

thf(4879,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1316]) ).

thf(484,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right1_4) ).

thf(2927,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(1713,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_lessI_0) ).

thf(5807,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1713]) ).

thf(1802,axiom,
    ! [A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_numeral__1__eq__1_0) ).

thf(6008,plain,
    ! [A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1802]) ).

thf(1288,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( J @ hAPP ) )
        = C )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__0__0_1) ).

thf(4812,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( J @ hAPP ) )
        = C )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1288]) ).

thf(1364,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( I @ ( H @ ( D @ ( G @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__ring__1_Opower__minus__even_0) ).

thf(4986,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( I @ ( H @ ( D @ ( G @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1364]) ).

thf(1716,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Oprod__diff__prod_0) ).

thf(5812,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1716]) ).

thf(1650,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_number__of__Bit1_0) ).

thf(5661,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1650]) ).

thf(1567,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( M @ ( F @ ( E @ ( L @ ( K @ ( C @ ( D @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__le__zero__iff_0) ).

thf(5463,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( M @ ( F @ ( E @ ( L @ ( K @ ( C @ ( D @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( G @ ( F @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1567]) ).

thf(741,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__mult__minus_0) ).

thf(3522,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(1085,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__eq__add__iff1_0) ).

thf(4331,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).

thf(107,axiom,
    tc_Int_Oint @ class_Int_Onumber,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Int_Onumber) ).

thf(2054,plain,
    tc_Int_Oint @ class_Int_Onumber,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(1125,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Omul__0_0) ).

thf(4426,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).

thf(1031,axiom,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__neq__Zero_0) ).

thf(4207,plain,
    ! [A: $i] :
      ( ( A @ ( c_Suc @ hAPP ) )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).

thf(1115,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__le__mult__iff_0) ).

thf(4405,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).

thf(479,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__antisym__conv1_0) ).

thf(2914,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(433,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_1) ).

thf(2816,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(1596,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__semiring__class_Odistrib_0) ).

thf(5538,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1596]) ).

thf(1618,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__cancel2_0) ).

thf(5587,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1618]) ).

thf(1001,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__strict__right__mono_0) ).

thf(4136,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(126,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Opordered__ring) ).

thf(2096,plain,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(370,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__less__iff__less_1) ).

thf(2665,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(1139,axiom,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_numeral__1__eq__Suc__0_0) ).

thf(4458,plain,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).

thf(763,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mod__eq__iff_1) ).

thf(3576,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(1055,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__le_0) ).

thf(4260,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).

thf(513,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__by__0_0) ).

thf(2994,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(1380,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Omul__pwr_0) ).

thf(5025,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1380]) ).

thf(419,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__divide_0) ).

thf(2784,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(200,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right1_4) ).

thf(2271,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( G @ ( K @ ( E @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( D @ ( G @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(607,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_8) ).

thf(3206,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(286,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__eq__mult_0) ).

thf(2471,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(1198,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( A @ ( K @ ( C @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( H = B )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__gt__zero__iff_2) ).

thf(4598,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( M @ ( E @ ( D @ ( L @ ( A @ ( K @ ( C @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( H = B )
      | ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).

thf(1691,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_trans__less__add1_0) ).

thf(5757,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1691]) ).

thf(100,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
        = ( B @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ring_Oneg__mul_0) ).

thf(2037,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
        = ( B @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(938,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I14_J_0) ).

thf(3981,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(462,axiom,
    tc_Int_Oint @ class_OrderedGroup_Opordered__comm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Opordered__comm__monoid__add) ).

thf(2878,plain,
    tc_Int_Oint @ class_OrderedGroup_Opordered__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(299,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__add__class_Oadd__0__right_0) ).

thf(2503,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(1068,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( L @ ( C @ ( B @ ( A @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Oabs__power2_0) ).

thf(4292,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( L @ ( C @ ( B @ ( A @ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(276,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__add__self2_0) ).

thf(2449,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(79,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_8) ).

thf(1990,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(1308,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__0__left_1) ).

thf(4859,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1308]) ).

thf(378,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__strict__increasing2_0) ).

thf(2684,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(1236,axiom,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ocomm__semiring__1) ).

thf(4688,plain,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1236]) ).

thf(227,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__divide__mult__cancel__right_0) ).

thf(2333,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(447,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__iff__diff__eq__0_1) ).

thf(2846,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(638,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( M @ ( C @ ( B @ ( A @ ( L @ ( K @ ( J @ ( I @ ( E @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( G @ ( F @ ( E @ hAPP ) @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( G @ ( F @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Oabs__power__minus_0) ).

thf(3278,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( M @ ( C @ ( B @ ( A @ ( L @ ( K @ ( J @ ( I @ ( E @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( G @ ( F @ ( E @ hAPP ) @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) )
        = ( G @ ( F @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(341,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_2) ).

thf(2597,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(1523,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__diff__left__eq_0) ).

thf(5358,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1523]) ).

thf(1251,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__Suc__diff__eq1_0) ).

thf(4722,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1251]) ).

thf(1611,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel2_1) ).

thf(5570,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1611]) ).

thf(1720,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__ge__zero_0) ).

thf(5822,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1720]) ).

thf(391,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__eq__iff_2) ).

thf(2715,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(790,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__eq__zero__iff_0) ).

thf(3640,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(907,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Oring__1) ).

thf(3913,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(565,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__divide__iff_2) ).

thf(3112,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(611,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__eq__mult_1) ).

thf(3214,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(903,axiom,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ocomm__semiring) ).

thf(3904,plain,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(1687,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_4) ).

thf(5748,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1687]) ).

thf(871,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__strict__mono_0) ).

thf(3833,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(1144,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( H @ ( G @ ( D @ ( F @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__ring__1_Opower__minus1__odd_0) ).

thf(4469,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( H @ ( G @ ( D @ ( F @ ( B @ ( A @ c_Ring__and__Field_Ocomm__ring__1 ) ) ) ) ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( D @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1144]) ).

thf(1358,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__gt1_0) ).

thf(4971,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1358]) ).

thf(941,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__diff__iff_0) ).

thf(3988,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(303,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left1_1) ).

thf(2513,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(12,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__ring__abs,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Opordered__ring__abs) ).

thf(1845,plain,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__ring__abs,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(1262,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__le__mono2_0) ).

thf(4749,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1262]) ).

thf(1053,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__iff1_1) ).

thf(4255,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).

thf(685,axiom,
    tc_Int_Oint @ class_OrderedGroup_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Ocomm__monoid__mult) ).

thf(3391,plain,
    tc_Int_Oint @ class_OrderedGroup_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(1635,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Odiff__left_0) ).

thf(5624,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1635]) ).

thf(1465,axiom,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__nat__eq_1) ).

thf(5229,plain,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1465]) ).

thf(64,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Opordered__semiring) ).

thf(1961,plain,
    tc_Int_Oint @ class_Ring__and__Field_Opordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(960,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I13_J_1) ).

thf(4035,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(1097,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel1_1) ).

thf(4360,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).

thf(1494,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__add_3) ).

thf(5294,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1494]) ).

thf(1764,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Ono__zero__divisors) ).

thf(5927,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1764]) ).

thf(1624,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__1_0) ).

thf(5602,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1624]) ).

thf(1342,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mod__eq__add3__mod__number__of_0) ).

thf(4937,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1342]) ).

thf(1740,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__product__nat_1) ).

thf(5871,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1740]) ).

thf(110,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__iff_3) ).

thf(2060,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(1343,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__1__left_0) ).

thf(4939,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1343]) ).

thf(553,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_6) ).

thf(3082,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(167,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__self_0) ).

thf(2192,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(1229,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__Suc__eq__le_0) ).

thf(4673,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1229]) ).

thf(1581,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( D @ ( C @ hAPP ) ) )
      | ~ ( K @ ( C @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osgn__0__0_0) ).

thf(5499,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( D @ ( C @ hAPP ) ) )
      | ~ ( K @ ( C @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1581]) ).

thf(1349,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Oab__semigroup__add) ).

thf(4950,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1349]) ).

thf(630,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_8) ).

thf(3260,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(337,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__mult__iff_5) ).

thf(2588,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(397,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left2_3) ).

thf(2732,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(1027,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( c_Suc @ hAPP ) @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( D @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div_H_7) ).

thf(4200,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( c_Suc @ hAPP ) @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( D @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).

thf(1789,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq__number__of_3) ).

thf(5980,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1789]) ).

thf(1590,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__mult_0) ).

thf(5523,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1590]) ).

thf(305,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = C )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__imp_0) ).

thf(2518,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = C )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(1165,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__diff_1) ).

thf(4517,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1165]) ).

thf(1216,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I8_J_0) ).

thf(4641,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ c_Int_OBit0 @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OMin @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1216]) ).

thf(1675,axiom,
    tc_nat @ class_Ring__and__Field_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Omult__zero) ).

thf(5721,plain,
    tc_nat @ class_Ring__and__Field_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1675]) ).

thf(160,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__eq__mult_3) ).

thf(2175,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(258,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left1_2) ).

thf(2407,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(851,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__0__left__number__of_0) ).

thf(3786,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(1403,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__nat__div__two__times__two_0) ).

thf(5079,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1403]) ).

thf(1211,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) )
        = B )
      | ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__Suc_1) ).

thf(4630,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) )
        = B )
      | ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1211]) ).

thf(440,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_4) ).

thf(2831,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(380,axiom,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__0__eq_1) ).

thf(2689,plain,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(552,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__linear_0) ).

thf(3080,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(1553,axiom,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( c_Suc @ c_COMBB ) ) ) ) @ c_Fun_Oinj__on ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__double__inj__on_0) ).

thf(5431,plain,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ ( c_Suc @ c_COMBB ) ) ) ) @ c_Fun_Oinj__on ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1553]) ).

thf(1758,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comp__arith_I115_J_0) ).

thf(5912,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1758]) ).

thf(1689,axiom,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocomm__monoid__mult) ).

thf(5753,plain,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1689]) ).

thf(1604,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_3) ).

thf(5555,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1604]) ).

thf(866,axiom,
    ( ( tc_nat @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__number__of__Min_0) ).

thf(3823,plain,
    ( ( tc_nat @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(1414,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( M @ ( C @ ( B @ ( L @ ( A @ ( K @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( G @ ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Oodd__power__less__zero_0) ).

thf(5106,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( M @ ( C @ ( B @ ( L @ ( A @ ( K @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( G @ ( F @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( E @ ( D @ ( C @ ( B @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1414]) ).

thf(844,axiom,
    tc_nat @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Power_Opower) ).

thf(3769,plain,
    tc_nat @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(1117,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__increasing__iff_1) ).

thf(4409,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).

thf(624,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_3) ).

thf(3246,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(1130,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__numeral__1_0) ).

thf(4437,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).

thf(1167,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I18_J_0) ).

thf(4522,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).

thf(1077,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__add_0) ).

thf(4314,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).

thf(1480,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Deriv_Oinverse__diff__inverse_0) ).

thf(5262,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1480]) ).

thf(1468,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__add_1) ).

thf(5235,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1468]) ).

thf(1695,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__eq__power2_1) ).

thf(5767,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1695]) ).

thf(984,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__less__Suc__eq_1) ).

thf(4092,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(1046,axiom,
    tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_pos2_0) ).

thf(4240,plain,
    tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).

thf(124,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I25_J_0) ).

thf(2092,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(78,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq_4) ).

thf(1987,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(1548,axiom,
    tc_nat @ class_Ring__and__Field_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ono__zero__divisors) ).

thf(5420,plain,
    tc_nat @ class_Ring__and__Field_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1548]) ).

thf(1813,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__neq__iff_0) ).

thf(6036,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1813]) ).

thf(1547,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_0) ).

thf(5418,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1547]) ).

thf(278,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__mult1__if_0) ).

thf(2454,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(1680,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I24_J_0) ).

thf(5731,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1680]) ).

thf(864,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right__le__imp__le_0) ).

thf(3818,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(566,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Osubr0__iff_0) ).

thf(3114,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(636,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__eq__zero__iff_2) ).

thf(3273,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(853,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__gt__zero__iff_2) ).

thf(3791,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(544,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__add1__eq_0) ).

thf(3063,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(481,axiom,
    tc_Int_Oint @ class_Int_Onumber__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Int_Onumber__ring) ).

thf(2920,plain,
    tc_Int_Oint @ class_Int_Onumber__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(824,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__difference__nat_3) ).

thf(3722,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(954,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Nat_Odiff__diff__eq_0) ).

thf(4020,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(796,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__abs_0) ).

thf(3655,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(1773,axiom,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__power2_0) ).

thf(5948,plain,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1773]) ).

thf(202,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__0__iff_2) ).

thf(2276,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(179,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_HOL_Oabs__class_Oabs ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_estimate__by__abs_0) ).

thf(2219,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_HOL_Oabs__class_Oabs ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(74,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__minus__le__zero_0) ).

thf(1980,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(1454,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A
        = ( A @ c_Parity_Osko__Parity__Xodd__Suc__mult__two__ex__1__1 @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__Suc__mult__two__ex_0) ).

thf(5203,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( A
        = ( A @ c_Parity_Osko__Parity__Xodd__Suc__mult__two__ex__1__1 @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1454]) ).

thf(1382,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__mono_0) ).

thf(5030,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1382]) ).

thf(857,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__imp__div__pos__less_0) ).

thf(3801,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(1501,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__imp__le_0) ).

thf(5309,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1501]) ).

thf(1797,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__eq_0) ).

thf(5995,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1797]) ).

thf(571,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__zero__imp__zero_0) ).

thf(3127,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(1370,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__is__0_0) ).

thf(5001,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1370]) ).

thf(385,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__right__mono_0) ).

thf(2701,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(1047,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__SucE_0) ).

thf(4241,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).

thf(319,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oplus__class_Oplus @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Ogb__semiring__axioms_H_0) ).

thf(2545,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oplus__class_Oplus @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(342,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of_1) ).

thf(2599,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
       != ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(671,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left2_4) ).

thf(3359,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( G @ ( K @ ( A @ ( E @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( G @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(659,axiom,
    tc_Int_Oint @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Opordered__cancel__ab__semigroup__add) ).

thf(3329,plain,
    tc_Int_Oint @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(103,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__not__less_1) ).

thf(2044,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(1446,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__numeral__1_0) ).

thf(5186,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1446]) ).

thf(1487,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq_5) ).

thf(5277,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1487]) ).

thf(820,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less0_0) ).

thf(3715,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(235,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ab__left__minus_0) ).

thf(2354,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(190,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__minus_0) ).

thf(2247,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(247,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left2_2) ).

thf(2379,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( K @ ( J @ ( A @ ( E @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( C @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(1028,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Osemiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Osemiring__1) ).

thf(4202,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Osemiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).

thf(1239,axiom,
    tc_Complex_Ocomplex @ class_Int_Oring__char__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Int_Oring__char__0) ).

thf(4695,plain,
    tc_Complex_Ocomplex @ class_Int_Oring__char__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1239]) ).

thf(1089,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comp__arith_I113_J_0) ).

thf(4342,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
    = ( tc_nat @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).

thf(1673,axiom,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ c_FFT__Mirabelle_Oroot )
    = c_Complex_Oii ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_root4_0) ).

thf(5717,plain,
    ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ c_FFT__Mirabelle_Oroot )
    = c_Complex_Oii ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1673]) ).

thf(1119,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__less__cancel1_0) ).

thf(4414,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).

thf(1036,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__0__number__of_0) ).

thf(4217,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).

thf(1456,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_root__cancel1_0) ).

thf(5208,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_FFT__Mirabelle_Oroot @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1456]) ).

thf(52,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__1_0) ).

thf(1934,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(813,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Ozero__neq__one) ).

thf(3697,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(260,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
       != ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__equal__iff__equal_0) ).

thf(2411,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) )
       != ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(728,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_lemma__not__even__div2_0) ).

thf(3488,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(1284,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I14_J_0) ).

thf(4802,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit0 @ ( A @ c_Int_OBit0 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1284]) ).

thf(1752,axiom,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ ( c_Suc @ c_Fun_Oinj__on ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inj__Suc_0) ).

thf(5902,plain,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ ( c_Suc @ c_Fun_Oinj__on ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1752]) ).

thf(970,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__odd__power_0) ).

thf(4057,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(37,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__neg__neg_0) ).

thf(1898,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(15,axiom,
    tc_Int_Oint @ class_OrderedGroup_Olordered__ab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Olordered__ab__group__add) ).

thf(1849,plain,
    tc_Int_Oint @ class_OrderedGroup_Olordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(795,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__cases_0) ).

thf(3652,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(325,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right1_1) ).

thf(2558,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(934,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( E @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( C @ ( A @ ( E @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__divides__special_0) ).

thf(3970,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( E @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( C @ ( A @ ( E @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(869,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__add__inverse_0) ).

thf(3829,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(1682,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( A @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__imp__diff__is__add_1) ).

thf(5736,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( A @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1682]) ).

thf(1000,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq__number__of_4) ).

thf(4133,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(512,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__triangle__ineq4_0) ).

thf(2992,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(1203,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__odd__eq_0) ).

thf(4611,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1203]) ).

thf(1360,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__is__0__eq_0) ).

thf(4976,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1360]) ).

thf(318,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__double__add__iff__zero__less__single__add_0) ).

thf(2543,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(670,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__iff__diff__eq__0_0) ).

thf(3356,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(1470,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_0) ).

thf(5240,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1470]) ).

thf(1162,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I37_J_0) ).

thf(4510,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).

thf(236,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__squares__le__zero__iff_2) ).

thf(2357,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(70,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__less__dividend_0) ).

thf(1972,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(389,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( F @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right2_0) ).

thf(2711,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( F @ ( F @ ( B @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(964,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I17_J_0) ).

thf(4046,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ c_Int_OBit1 @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(1174,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__less__0_0) ).

thf(4538,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).

thf(567,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__le__cancel__left_1) ).

thf(3117,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(454,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__one__le__zero_0) ).

thf(2860,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(930,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__le__mono_0) ).

thf(3961,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(1570,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I13_J_0) ).

thf(5472,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OMin @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1570]) ).

thf(30,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__le__mono2_0) ).

thf(1881,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(908,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__even__power_H_0) ).

thf(3914,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(1411,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Ocomm__semiring) ).

thf(5100,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1411]) ).

thf(1281,axiom,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocomm__monoid__add) ).

thf(4795,plain,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1281]) ).

thf(1621,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__less__Suc_0) ).

thf(5596,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1621]) ).

thf(674,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Osubr0__iff_1) ).

thf(3366,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(519,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mod__equality2_0) ).

thf(3007,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(198,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_7) ).

thf(2266,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(335,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
      = ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__add__num__eq__if2__nat_0) ).

thf(2584,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
      = ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(1054,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( C @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Omult__1__right_0) ).

thf(4257,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( C @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(1228,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__gr__0_1) ).

thf(4671,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1228]) ).

thf(1291,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring) ).

thf(4819,plain,
    tc_nat @ class_Ring__and__Field_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1291]) ).

thf(808,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__eq__add__iff2_1) ).

thf(3684,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

thf(643,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_arith__simps_I31_J_0) ).

thf(3288,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( c_Int_OMin @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(769,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I1_J_1) ).

thf(3589,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(67,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__pos__nonneg_0) ).

thf(1966,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(709,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_7) ).

thf(3447,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(223,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gt__half__sum_0) ).

thf(2325,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(175,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__le__self__iff_1) ).

thf(2209,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(456,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__0__iff_5) ).

thf(2865,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(1724,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( G
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) )
      | ~ ( F @ ( L @ ( E @ ( D @ ( K @ ( J @ ( I @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__eq__zero__iff_0) ).

thf(5832,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( G
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) )
      | ~ ( F @ ( L @ ( E @ ( D @ ( K @ ( J @ ( I @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( H @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( G @ ( F @ ( E @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1724]) ).

thf(590,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__minus__eq_1) ).

thf(3168,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(55,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__self2_0) ).

thf(1940,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(269,axiom,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Parity_Oeven__odd__class_Oeven ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__zero__nat_0) ).

thf(2435,plain,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Parity_Oeven__odd__class_Oeven ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(1215,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__less__imp__less_0) ).

thf(4639,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1215]) ).

thf(640,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__left__mono_0) ).

thf(3284,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(89,axiom,
    tc_Int_Oint @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Ocancel__ab__semigroup__add) ).

thf(2013,plain,
    tc_Int_Oint @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(483,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__add__left__eq_0) ).

thf(2924,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(723,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__abs__iff_0) ).

thf(3478,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(1769,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__Suc__less__one_0) ).

thf(5939,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1769]) ).

thf(1067,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I19_J_0) ).

thf(4289,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(1063,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult1__eq_0) ).

thf(4278,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(576,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__right2_3) ).

thf(3139,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( D @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(1741,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__2__eq__Suc_H_0) ).

thf(5873,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1741]) ).

thf(1577,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__less__eq_0) ).

thf(5490,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1577]) ).

thf(10,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__eq__0_0) ).

thf(1839,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(767,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__Suc_0) ).

thf(3584,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(711,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__minus__iff_0) ).

thf(3452,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(280,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__add__le__zero__iff__single__add__le__zero_1) ).

thf(2458,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(1002,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__if_1) ).

thf(4138,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(1241,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Oadd__right_0) ).

thf(4698,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1241]) ).

thf(289,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__nonpositive__iff__nonpositive_1) ).

thf(2479,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(1305,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div__lemma_0) ).

thf(4851,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1305]) ).

thf(1201,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I3_J_1) ).

thf(4606,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).

thf(690,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__neg__iff__add__eq__0_1) ).

thf(3401,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(1253,axiom,
    ( ( tc_nat @ c_HOL_Oone__class_Oone )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_One__nat__def_0) ).

thf(4727,plain,
    ( ( tc_nat @ c_HOL_Oone__class_Oone )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1253]) ).

thf(1760,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__eq__self__zero_0) ).

thf(5917,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1760]) ).

thf(1346,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__diff__iff_1) ).

thf(4945,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1346]) ).

thf(434,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__not__le_0) ).

thf(2818,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(547,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Oabs__if ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__if_0) ).

thf(3067,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Oabs__if ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(458,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nonpos__nonpos_0) ).

thf(2870,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(1013,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__diff_0) ).

thf(4165,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).

thf(439,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( D = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Oadd__scale__eq__noteq_0) ).

thf(2828,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( D = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(1595,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__left__cancel__less_1) ).

thf(5536,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1595]) ).

thf(1804,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__difference__nat_2) ).

thf(6013,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1804]) ).

thf(591,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_3) ).

thf(3171,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(1699,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__diff__1_0) ).

thf(5777,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1699]) ).

thf(1345,axiom,
    tc_nat @ class_OrderedGroup_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocancel__semigroup__add) ).

thf(4944,plain,
    tc_nat @ class_OrderedGroup_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1345]) ).

thf(1313,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__cube_0) ).

thf(4873,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1313]) ).

thf(118,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__imp__div__pos__le_0) ).

thf(2078,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(43,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_1) ).

thf(1913,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(1222,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__eq__less__or__eq_0) ).

thf(4656,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1222]) ).

thf(217,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__mult__pos_0) ).

thf(2310,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(1613,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_left__distrib__number__of_0) ).

thf(5573,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1613]) ).

thf(598,axiom,
    tc_Int_Oint @ class_OrderedGroup_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Omonoid__add) ).

thf(3188,plain,
    tc_Int_Oint @ class_OrderedGroup_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(1585,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I35_J_0) ).

thf(5510,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1585]) ).

thf(1183,axiom,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Osum__power2__eq__zero__iff_2) ).

thf(4559,plain,
    ! [K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( K @ ( I @ ( H @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( K @ ( J @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).

thf(1767,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Opwr__0_0) ).

thf(5934,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1767]) ).

thf(1549,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I12_J_1) ).

thf(5421,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_Int_OBit1 @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1549]) ).

thf(978,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__if_0) ).

thf(4078,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(1756,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__left__eq_0) ).

thf(5907,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1756]) ).

thf(514,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( E @ c_lessequals ) ) )
      | ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_frac__le_0) ).

thf(2997,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( E @ c_lessequals ) ) )
      | ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(988,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__distrib2_0) ).

thf(4102,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(666,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_5) ).

thf(3346,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(949,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__self__div__2_0) ).

thf(4006,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(656,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__self1_0) ).

thf(3320,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(215,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_2) ).

thf(2305,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(1422,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__le__add__iff2_0) ).

thf(5124,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1422]) ).

thf(1015,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__neg_1) ).

thf(4170,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).

thf(231,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I2_J_0) ).

thf(2342,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(541,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__Suc_0) ).

thf(3056,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(1575,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq_4) ).

thf(5485,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1575]) ).

thf(1511,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__less_0) ).

thf(5332,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1511]) ).

thf(23,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__right__mono__neg_0) ).

thf(1866,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(114,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__iff_3) ).

thf(2071,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(367,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( F @ ( L @ ( A @ ( K @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( H @ ( G @ ( F @ hAPP ) @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Ozero__le__power__abs_0) ).

thf(2658,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( M @ ( D @ ( C @ ( F @ ( L @ ( A @ ( K @ ( B @ ( J @ ( I @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( H @ ( G @ ( F @ hAPP ) @ ( E @ ( D @ ( C @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(915,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Nat_Oadd__0__right_0) ).

thf(3929,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(1399,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult2__eq_0) ).

thf(5068,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1399]) ).

thf(475,axiom,
    tc_Int_Oint @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Divides_Osemiring__div) ).

thf(2907,plain,
    tc_Int_Oint @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(1075,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__eq__Suc__le_0) ).

thf(4310,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(422,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( E @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right2_1) ).

thf(2792,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( E @ ( C @ ( G @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(800,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( D @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__eq__less_0) ).

thf(3665,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( D @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(1232,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I1_J_0) ).

thf(4678,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( B @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1232]) ).

thf(82,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__le_2) ).

thf(1998,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(1188,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__less__imp__less__base_0) ).

thf(4574,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).

thf(855,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of1_1) ).

thf(3797,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(771,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult_Ozero__right_0) ).

thf(3593,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(1331,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Onoteq__reduce_1) ).

thf(4913,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1331]) ).

thf(201,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__add__right__eq_0) ).

thf(2273,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(852,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq__number__of_7) ).

thf(3789,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(867,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__le__zero__eq_2) ).

thf(3825,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(681,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__le__iff_0) ).

thf(3383,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(1051,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( B @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__mul_0) ).

thf(4250,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( B @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).

thf(18,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__minus__commute_0) ).

thf(1854,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(1317,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__neg_0) ).

thf(4881,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1317]) ).

thf(1762,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__strict__right__mono_0) ).

thf(5922,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1762]) ).

thf(1638,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__nat__equiv__def_0) ).

thf(5632,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1638]) ).

thf(273,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__1__iff_1) ).

thf(2443,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(827,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__idem_0) ).

thf(3729,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(995,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__diff__right_0) ).

thf(4121,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(1246,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__1__iff_2) ).

thf(4710,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1246]) ).

thf(510,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__power__divide_0) ).

thf(2988,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(885,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_2) ).

thf(3864,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(205,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( E @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_frac__less2_0) ).

thf(2283,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( E @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( E @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(251,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_xt1_I7_J_0) ).

thf(2389,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(1286,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__number__of__mult__left_0) ).

thf(4806,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1286]) ).

thf(39,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( D @ ( A @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__one_0) ).

thf(1903,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( D @ ( A @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(1226,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_square__eq__iff_1) ).

thf(4666,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1226]) ).

thf(143,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of1_4) ).

thf(2135,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(1533,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of1_7) ).

thf(5382,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1533]) ).

thf(323,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__pos__neg_0) ).

thf(2554,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(312,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__add__one_0) ).

thf(2532,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(1448,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel1_1) ).

thf(5191,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1448]) ).

thf(1709,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__pos__neg_0) ).

thf(5800,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1709]) ).

thf(1425,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__le__mono1_0) ).

thf(5132,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1425]) ).

thf(662,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_4) ).

thf(3336,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(1751,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__numeral__0_0) ).

thf(5899,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1751]) ).

thf(675,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__abs__divide_0) ).

thf(3369,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(997,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel2_2) ).

thf(4126,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(613,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__zero_0) ).

thf(3220,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(1176,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__div__lemma_2) ).

thf(4542,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).

thf(1248,axiom,
    tc_nat @ class_Ring__and__Field_Omult__mono1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Omult__mono1) ).

thf(4715,plain,
    tc_nat @ class_Ring__and__Field_Omult__mono1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1248]) ).

thf(84,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_7) ).

thf(2004,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(95,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_6) ).

thf(2026,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(786,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__self__eq__0_0) ).

thf(3630,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(733,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__1__eq__mult__iff_0) ).

thf(3501,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(2,negated_conjecture,
    tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_conjecture_1) ).

thf(1822,plain,
    tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( v_ka____ @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( v_i____ @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(1298,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__add2_0) ).

thf(4834,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1298]) ).

thf(739,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__neqE_0) ).

thf(3516,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(1155,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__iff__Suc__add_1) ).

thf(4495,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).

thf(1244,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__n__div__2__gt__zero_0) ).

thf(4705,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1244]) ).

thf(141,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__divide__iff_5) ).

thf(2130,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(883,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ( A @ ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_iszero__number__of__Bit0_1) ).

thf(3859,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ( A @ ( A @ ( B @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ c_Int_Oiszero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Int_Oring__char__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(627,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__0__iff_0) ).

thf(3252,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(1583,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__def_0) ).

thf(5505,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1583]) ).

thf(1325,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__power__nat_0) ).

thf(4900,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1325]) ).

thf(196,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__add__eq_0) ).

thf(2262,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(21,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_eq__divide__eq__number__of_3) ).

thf(1861,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ ( C @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(249,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__add__distrib_0) ).

thf(2384,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(1026,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Oadd__cancel_0) ).

thf(4197,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(1543,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_plus__nat_Oadd__0_0) ).

thf(5408,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1543]) ).

thf(1008,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I9_J_0) ).

thf(4153,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).

thf(1806,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__mult__right_0) ).

thf(6018,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1806]) ).

thf(1683,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__mod__geq_0) ).

thf(5739,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1683]) ).

thf(1737,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( c_Suc @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power_Opower_Opower__Suc_0) ).

thf(5864,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( c_Suc @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1737]) ).

thf(1260,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_3) ).

thf(4745,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1260]) ).

thf(275,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__le__cancel__left_0) ).

thf(2447,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(491,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__eq__divide__number__of_0) ).

thf(2944,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(1780,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Odivision__by__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Odivision__by__zero) ).

thf(5963,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Odivision__by__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1780]) ).

thf(495,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq_4) ).

thf(2956,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(424,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__less__0__iff_1) ).

thf(2796,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(1437,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__mult__commute_0) ).

thf(5164,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1437]) ).

thf(478,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_2) ).

thf(2912,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(1080,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_trans__le__add2_0) ).

thf(4321,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).

thf(255,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zmod__simps_I1_J_0) ).

thf(2399,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(702,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_2) ).

thf(3430,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(1195,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__power__if_0) ).

thf(4591,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).

thf(1273,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__mult__self_0) ).

thf(4774,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1273]) ).

thf(773,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Omonoid__mult) ).

thf(3598,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(282,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__inverse__eq_1) ).

thf(2462,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(1379,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__iff__diff__less__0_0) ).

thf(5023,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1379]) ).

thf(581,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__less__iff_0) ).

thf(3149,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(226,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_right__inverse_0) ).

thf(2330,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(900,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Opwr__mul_0) ).

thf(3896,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(170,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_8) ).

thf(2198,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(1109,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( D @ ( F @ ( E @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( D @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I4_J_0) ).

thf(4387,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( D @ ( F @ ( E @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( D @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).

thf(332,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( D @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( C @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( B @ ( D @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Rational_Oordered__idom_Osgn__less_0) ).

thf(2576,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( D @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( C @ ( F @ ( E @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( B @ ( D @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(805,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__nat__eq_2) ).

thf(3677,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(238,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_9) ).

thf(2362,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(1601,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__assoc_0) ).

thf(5549,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1601]) ).

thf(1610,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
      = ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__plus__one__nat_0) ).

thf(5568,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
      = ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1610]) ).

thf(668,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_7) ).

thf(3351,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(1739,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__Suc_0) ).

thf(5869,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1739]) ).

thf(943,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__2__eq__Suc_0) ).

thf(3993,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(473,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__one_0) ).

thf(2902,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(137,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_8) ).

thf(2120,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(1420,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__2__cases_0) ).

thf(5119,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1420]) ).

thf(125,axiom,
    tc_Int_Oint @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Orderings_Olinorder) ).

thf(2095,plain,
    tc_Int_Oint @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(1619,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__eq__square__number__of_0) ).

thf(5590,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1619]) ).

thf(184,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_field__inverse_0) ).

thf(2230,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(35,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__le__inverse__iff_2) ).

thf(1893,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(210,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__le__less_0) ).

thf(2294,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(295,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left1_2) ).

thf(2494,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( K @ ( J @ ( E @ ( A @ ( I @ ( F @ ( H @ ( G @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( F @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( F @ ( B @ ( E @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(1450,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel__left__neg_1) ).

thf(5196,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1450]) ).

thf(480,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__of__neg_0) ).

thf(2917,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( B @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(1102,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel2_0) ).

thf(4372,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).

thf(840,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__Suc__eq__le_1) ).

thf(3761,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(1073,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_trans__le__add1_0) ).

thf(4306,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(1667,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__asym_H_0) ).

thf(5704,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1667]) ).

thf(1513,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__left__pos_1) ).

thf(5336,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1513]) ).

thf(417,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of_1) ).

thf(2779,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(92,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__left__mono__neg_0) ).

thf(2019,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(1207,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__eq_0) ).

thf(4619,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1207]) ).

thf(244,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__mult__iff_1) ).

thf(2374,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(207,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__nonpos__pos_0) ).

thf(2287,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(1333,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__eq__number__of_5) ).

thf(4917,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( tc_nat @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1333]) ).

thf(1373,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__le__zero__iff_0) ).

thf(5009,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1373]) ).

thf(1678,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__power__nat__eq__number__of_2) ).

thf(5728,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1678]) ).

thf(1371,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__nat__number__of_0) ).

thf(5004,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1371]) ).

thf(1197,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__le_1) ).

thf(4596,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).

thf(1250,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__cancel1_0) ).

thf(4719,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1250]) ).

thf(1694,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I16_J_0) ).

thf(5764,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1694]) ).

thf(1572,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__0_1) ).

thf(5477,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1572]) ).

thf(1137,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_6) ).

thf(4455,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).

thf(41,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__minus__self__iff_1) ).

thf(1908,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(146,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__less__inverse__iff_1) ).

thf(2142,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(679,axiom,
    tc_Int_Oint @ class_OrderedGroup_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Ocomm__monoid__add) ).

thf(3379,plain,
    tc_Int_Oint @ class_OrderedGroup_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(797,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__right_Ozero_0) ).

thf(3658,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_RealVector_Oreal__normed__algebra ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(194,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__num__frac_0) ).

thf(2258,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(1391,axiom,
    tc_nat @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add) ).

thf(5051,plain,
    tc_nat @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1391]) ).

thf(952,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__1_0) ).

thf(4014,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(1424,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Onoteq__reduce_0) ).

thf(5129,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1424]) ).

thf(1145,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__le__mono_0) ).

thf(4472,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).

thf(1113,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__even__eq_0) ).

thf(4399,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).

thf(737,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__minus1__odd_0) ).

thf(3511,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(1626,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBB ) ) ) ) @ hAPP ) )
      = ( F @ ( B @ hAPP ) @ ( A @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_COMBB__def_0) ).

thf(5606,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBB ) ) ) ) @ hAPP ) )
      = ( F @ ( B @ hAPP ) @ ( A @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1626]) ).

thf(1105,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__cong_0) ).

thf(4378,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).

thf(313,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left1_0) ).

thf(2534,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( B @ ( K @ ( D @ ( A @ ( J @ ( E @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( C @ ( F @ ( G @ hAPP ) @ hAPP ) @ ( F @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(193,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = B )
      | ( ( A @ c_HOL_Ozero__class_Ozero )
       != ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__0__equal__iff__equal_0) ).

thf(2255,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = B )
      | ( ( A @ c_HOL_Ozero__class_Ozero )
       != ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(644,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mod__equality_0) ).

thf(3291,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(1095,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__mono__odd_0) ).

thf(4356,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).

thf(1707,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__sum__nat_1) ).

thf(5796,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1707]) ).

thf(1287,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Deriv_Oadd__diff__add_0) ).

thf(4809,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1287]) ).

thf(508,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_6) ).

thf(2983,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(775,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__1__add__number__of_0) ).

thf(3601,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( c_Int_OPls @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ c_HOL_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(185,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_2) ).

thf(2233,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(893,axiom,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__add__less1_0) ).

thf(3882,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(1776,axiom,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neq0__conv_1) ).

thf(5954,plain,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1776]) ).

thf(1467,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__power__if_1) ).

thf(5232,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( C @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1467]) ).

thf(1522,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__gt__zero__iff_0) ).

thf(5356,plain,
    ! [A: $i] :
      ( ~ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1522]) ).

thf(344,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Ocomm__semiring__1) ).

thf(2604,plain,
    tc_Int_Oint @ class_Ring__and__Field_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(1302,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel__left__neg_0) ).

thf(4843,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1302]) ).

thf(28,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_2) ).

thf(1876,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(399,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left__le__one__le_0) ).

thf(2736,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(1278,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__le__zero__iff_1) ).

thf(4787,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1278]) ).

thf(86,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__abs_0) ).

thf(2008,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(77,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Ocomm__ring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Ocomm__ring__1) ).

thf(1986,plain,
    tc_Int_Oint @ class_Ring__and__Field_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(1153,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__is__add_1) ).

thf(4491,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).

thf(731,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__less__add__iff2_0) ).

thf(3496,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(1190,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__div__geq_0) ).

thf(4579,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).

thf(1319,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__mono1_0) ).

thf(4885,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1319]) ).

thf(1375,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__mono1_0) ).

thf(5014,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1375]) ).

thf(810,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__if_1) ).

thf(3690,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(1539,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I35_J_0) ).

thf(5398,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( c_Suc @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1539]) ).

thf(1100,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_7) ).

thf(4367,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).

thf(1387,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__eq__add__iff1_1) ).

thf(5041,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1387]) ).

thf(588,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_negative__imp__inverse__negative_0) ).

thf(3164,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(1665,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__trans_0) ).

thf(5700,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1665]) ).

thf(927,axiom,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ c_Fun_Oinj__on ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_double__inj__on_0) ).

thf(3956,plain,
    ! [A: $i] : ( tc_nat @ ( tc_nat @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ c_Fun_Oinj__on ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(1497,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_termination__basic__simps_I3_J_0) ).

thf(5300,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1497]) ).

thf(879,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring__1) ).

thf(3850,plain,
    tc_nat @ class_Ring__and__Field_Osemiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(497,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__triangle__ineq2_0) ).

thf(2961,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( A @ ( C @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(181,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__double__add__iff__zero__le__single__add_0) ).

thf(2223,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(101,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_6) ).

thf(2040,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(1606,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__1_5) ).

thf(5559,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1606]) ).

thf(579,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__mono_0) ).

thf(3145,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(20,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_2) ).

thf(1859,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(1005,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__is__1_1) ).

thf(4146,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).

thf(957,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Oab__semigroup__mult) ).

thf(4029,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(1019,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__power2_0) ).

thf(4178,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oabs__class_Oabs ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).

thf(1072,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__right__eq_0) ).

thf(4303,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(1185,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( C = D )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Oadd__scale__eq__noteq_0) ).

thf(4565,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( C = D )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).

thf(1274,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__cancel1_1) ).

thf(4777,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1274]) ).

thf(1435,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__add__left__commute_0) ).

thf(5159,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1435]) ).

thf(1092,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__is__add_4) ).

thf(4350,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).

thf(1265,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I13_J_0) ).

thf(4755,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1265]) ).

thf(1408,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__1_0) ).

thf(5092,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1408]) ).

thf(757,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( F @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__pwr_0) ).

thf(3560,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( F @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(1669,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq__number__of_4) ).

thf(5709,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1669]) ).

thf(1066,axiom,
    c_Int_OPls != c_Int_OMin,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I37_J_0) ).

thf(4286,plain,
    c_Int_OPls != c_Int_OMin,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(756,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_4) ).

thf(3558,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(1795,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__left__idem_0) ).

thf(5990,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1795]) ).

thf(1032,axiom,
    tc_nat @ class_Ring__and__Field_Opordered__cancel__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Opordered__cancel__semiring) ).

thf(4210,plain,
    tc_nat @ class_Ring__and__Field_Opordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).

thf(890,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__diff__add__eq2_0) ).

thf(3875,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(1146,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__nonpos__nonneg_0) ).

thf(4474,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).

thf(1357,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__eq__0__iff__number__of_2) ).

thf(4968,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1357]) ).

thf(403,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_div__mult__mult1_0) ).

thf(2746,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(1377,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__neq__iff_1) ).

thf(5019,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1377]) ).

thf(1725,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I17_J_0) ).

thf(5835,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1725]) ).

thf(1608,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_FFT__Mirabelle_OFFT ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_FFT_Osimps_I1_J_0) ).

thf(5564,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_FFT__Mirabelle_OFFT ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1608]) ).

thf(262,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right1_3) ).

thf(2417,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( F @ ( A @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(240,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Omult__mono1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Omult__mono1) ).

thf(2366,plain,
    tc_Int_Oint @ class_Ring__and__Field_Omult__mono1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(1515,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( F @ ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( E @ ( A @ ( F @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( F @ ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_DERIV__mult__lemma_0) ).

thf(5339,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( F @ ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( E @ ( A @ ( F @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( F @ ( A @ ( E @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_RealVector_Oreal__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1515]) ).

thf(257,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_minus__add_0) ).

thf(2404,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(657,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__antisym__conv2_0) ).

thf(3323,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(700,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__0__1__iff_1) ).

thf(3426,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(1474,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__strict__increasing__iff_0) ).

thf(5251,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1474]) ).

thf(34,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__neg__neg_0) ).

thf(1891,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(1385,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_half__gt__zero_0) ).

thf(5036,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1385]) ).

thf(1310,axiom,
    ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_rel__simps_I3_J_0) ).

thf(4865,plain,
    ~ ( tc_Int_Oint @ ( c_Int_OMin @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1310]) ).

thf(1531,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Omult__zero) ).

thf(5378,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1531]) ).

thf(842,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__leD_0) ).

thf(3765,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(1642,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__add_1) ).

thf(5642,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1642]) ).

thf(351,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__less__eq_1) ).

thf(2619,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ ( C @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(699,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__nonneg__nonneg_0) ).

thf(3424,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(735,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__add__cancel_0) ).

thf(3506,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(876,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( C @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( c_Suc @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I27_J_0) ).

thf(3843,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( C @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( c_Suc @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(1452,axiom,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add) ).

thf(5200,plain,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1452]) ).

thf(803,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_half__gt__zero__iff_0) ).

thf(3673,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(93,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__eq__1__iff_1) ).

thf(2021,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(1416,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__Suc__mult__two__ex_1) ).

thf(5110,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1416]) ).

thf(1557,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__zero__right_0) ).

thf(5440,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1557]) ).

thf(364,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__1__iff_0) ).

thf(2651,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(623,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of_9) ).

thf(3244,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(6,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__minus__cancel_0) ).

thf(1829,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_HOL_Oabs__class_Oabs ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(1432,axiom,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__0__number__of_1) ).

thf(5151,plain,
    ! [A: $i] :
      ( ~ ( tc_Int_Oint @ ( A @ ( c_Int_OPls @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( A @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1432]) ).

thf(1150,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__number__of__eq__not__less_1) ).

thf(4484,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).

thf(932,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__neq__implies__less_0) ).

thf(3965,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(895,axiom,
    tc_nat @ class_Orderings_Oorder,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Orderings_Oorder) ).

thf(3886,plain,
    tc_nat @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(1352,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__nat__equiv__def2_1) ).

thf(4955,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1352]) ).

thf(1158,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__comm__semiring__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__comm__semiring__strict) ).

thf(4501,plain,
    tc_nat @ class_Ring__and__Field_Oordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).

thf(1060,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__add__inverse2_0) ).

thf(4271,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(144,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__antisym__conv2_1) ).

thf(2138,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(50,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__0__iff_4) ).

thf(1930,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(1486,axiom,
    ( ( tc_nat @ c_HOL_Oone__class_Oone )
    = ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__1__eq__mult__iff_2) ).

thf(5275,plain,
    ( ( tc_nat @ c_HOL_Oone__class_Oone )
    = ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1486]) ).

thf(140,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__zero_0) ).

thf(2127,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(1304,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__diff__inverse_0) ).

thf(4848,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1304]) ).

thf(153,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__increasing2_0) ).

thf(2159,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(431,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( F @ ( E @ ( D @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C = B )
      | ( B @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Olinorder__neqE__ordered__idom_0) ).

thf(2810,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( K @ ( J @ ( I @ ( A @ ( H @ ( G @ ( F @ ( E @ ( D @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( C = B )
      | ( B @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(527,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zmod__simps_I2_J_0) ).

thf(3025,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(5,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_abs__ge__self_0) ).

thf(1827,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(267,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod__mult__self2__is__0_0) ).

thf(2430,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(750,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__minus1__even_0) ).

thf(3543,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Ouminus__class_Ouminus ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(427,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__le__power__iff_1) ).

thf(2803,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(1283,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power3__eq__cube_0) ).

thf(4799,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1283]) ).

thf(471,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_split__mult__pos__le_0) ).

thf(2898,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(694,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__right2_5) ).

thf(3411,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(1428,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__num__iff_1) ).

thf(5140,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1428]) ).

thf(1640,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__add_2) ).

thf(5637,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1640]) ).

thf(197,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Osemiring__0) ).

thf(2265,plain,
    tc_Int_Oint @ class_Ring__and__Field_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(994,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__add__num__eq__if2__nat_1) ).

thf(4118,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Fact_Ofact__class_Ofact ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(1616,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__add__one__is__two_0) ).

thf(5581,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1616]) ).

thf(284,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__divide__eq_5) ).

thf(2467,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(1396,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__strict__left__mono_0) ).

thf(5061,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1396]) ).

thf(72,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__number__of_0) ).

thf(1976,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(1787,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__power__nat_1) ).

thf(5975,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1787]) ).

thf(1045,axiom,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring__1__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__Ring__and__Field_Oring__1__no__zero__divisors) ).

thf(4239,plain,
    tc_Complex_Ocomplex @ class_Ring__and__Field_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).

thf(1297,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_monoid__mult_Opower__even__eq_0) ).

thf(4831,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( A @ c_OrderedGroup_Omonoid__mult ) ) )
      | ( ( E @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1297]) ).

thf(686,axiom,
    tc_Int_Oint @ class_OrderedGroup_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Oab__semigroup__add) ).

thf(3392,plain,
    tc_Int_Oint @ class_OrderedGroup_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(1634,axiom,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Complex__Ocomplex__OrderedGroup_Ocomm__monoid__mult) ).

thf(5623,plain,
    tc_Complex_Ocomplex @ class_OrderedGroup_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1634]) ).

thf(1041,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mod2__Suc__Suc_0) ).

thf(4229,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).

thf(1389,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_one__is__add_2) ).

thf(5046,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1389]) ).

thf(1030,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__semiring__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__semiring__strict) ).

thf(4206,plain,
    tc_nat @ class_Ring__and__Field_Oordered__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(1698,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__eq__1__iff_0) ).

thf(5774,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1698]) ).

thf(1223,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__antisym_0) ).

thf(4659,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1223]) ).

thf(48,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_DERIV__inverse__lemma_0) ).

thf(1925,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) )
        = ( A @ ( A @ ( C @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_RealVector_Oreal__normed__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(1460,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__0_0) ).

thf(5219,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1460]) ).

thf(920,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_odd__power__less__zero_0) ).

thf(3939,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(316,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_xt1_I6_J_0) ).

thf(2539,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(1748,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_diff__Suc__less_0) ).

thf(5891,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1748]) ).

thf(1521,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__SucI_0) ).

thf(5354,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1521]) ).

thf(291,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__eq__eq__number__of1_0) ).

thf(2484,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(730,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__pos__pos_0) ).

thf(3494,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(1038,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__strict__decreasing_0) ).

thf(4221,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).

thf(243,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_3) ).

thf(2372,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( A @ ( D @ c_Int_Onumber__class_Onumber__of ) @ ( C @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(463,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__right__mono_0) ).

thf(2879,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(765,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_no__zero__divisors_0) ).

thf(3580,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(1401,axiom,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( K @ ( A @ ( J @ ( B @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( C @ ( B @ hAPP ) @ ( C @ ( B @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Oabs__mult__self_0) ).

thf(5073,plain,
    ! [L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( L @ ( K @ ( A @ ( J @ ( B @ ( I @ ( H @ ( G @ ( F @ ( E @ ( D @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( ( C @ ( B @ hAPP ) @ ( C @ ( B @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1401]) ).

thf(1270,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ ( D @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__mono__even_0) ).

thf(4767,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ c_Parity_Oeven__odd__class_Oeven ) )
      | ~ ( A @ ( A @ ( D @ c_HOL_Oabs__class_Oabs ) @ ( A @ ( B @ c_HOL_Oabs__class_Oabs ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1270]) ).

thf(14,axiom,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__group__add__abs,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__OrderedGroup_Opordered__ab__group__add__abs) ).

thf(1848,plain,
    tc_Int_Oint @ class_OrderedGroup_Opordered__ab__group__add__abs,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(703,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left1_3) ).

thf(3432,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( C @ ( E @ ( K @ ( A @ ( F @ ( J @ ( G @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ( B @ ( G @ ( F @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(539,axiom,
    tc_Int_Oint @ class_Ring__and__Field_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_Int__Oint__Ring__and__Field_Ozero__neq__one) ).

thf(3052,plain,
    tc_Int_Oint @ class_Ring__and__Field_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(537,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__number__of_1) ).

thf(3047,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Int_Oint @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(1017,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__Suc__less_0) ).

thf(4174,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(837,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zmod__simps_I4_J_0) ).

thf(3753,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(929,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__Suc__eq_2) ).

thf(3960,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(744,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_linorder__neqE__ordered__idom_0) ).

thf(3528,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(1441,axiom,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
        = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__Suc__mult__two__diff__two_0) ).

thf(5172,plain,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) )
        = ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1441]) ).

thf(1295,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_not__less__eq__eq_1) ).

thf(4826,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( c_Suc @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1295]) ).

thf(1771,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I2_J_0) ).

thf(5943,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1771]) ).

thf(705,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
       != ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Oadd__r0__iff_0) ).

thf(3437,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
       != ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(1275,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_natgb_Oadd__mul__solve_0) ).

thf(4779,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1275]) ).

thf(221,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__divide__eq__number__of1_2) ).

thf(2321,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ c_HOL_Oinverse__class_Odivide ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(616,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__frac__num_0) ).

thf(3227,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(1598,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( B @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( c_Suc @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I28_J_0) ).

thf(5542,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( B @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( c_Suc @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1598]) ).

thf(1210,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( B = C )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power2__eq__imp__eq_0) ).

thf(4627,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( B = C )
      | ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1210]) ).

thf(1192,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__strict__mono_H_0) ).

thf(4584,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).

thf(65,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq_0) ).

thf(1962,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(464,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__cancel__21_1) ).

thf(2881,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(1792,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ c_Int_OBit1 @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__special_I4_J_0) ).

thf(5986,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ c_Int_Onumber__class_Onumber__of ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_Int_Oint @ ( c_Int_OPls @ c_Int_OBit1 @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1792]) ).

thf(57,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_zero__less__two_0) ).

thf(1945,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(610,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__le__cancel__left2_5) ).

thf(3212,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( B @ ( E @ ( K @ ( J @ ( A @ ( I @ ( H @ ( G @ ( F @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(1645,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__mod__eq__add3__mod_0) ).

thf(5649,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ ( c_Suc @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1645]) ).

thf(1565,axiom,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__eq__zero__iff_2) ).

thf(5458,plain,
    ! [A: $i] :
      ( ( ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1565]) ).

thf(1704,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_sum__power2__gt__zero__iff_1) ).

thf(5789,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1704]) ).

thf(1632,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_add__Suc__right_0) ).

thf(5618,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( c_Suc @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1632]) ).

thf(1277,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I7_J_0) ).

thf(4784,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1277]) ).

thf(559,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__le__0__iff__le_0) ).

thf(3096,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(881,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_lemma__odd__mod__4__div__2_0) ).

thf(3854,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(381,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_le__iff__diff__le__0_1) ).

thf(2690,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(1445,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I7_J_0) ).

thf(5183,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1445]) ).

thf(1479,axiom,
    tc_nat @ class_OrderedGroup_Opordered__comm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__comm__monoid__add) ).

thf(5261,plain,
    tc_nat @ class_OrderedGroup_Opordered__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1479]) ).

thf(1178,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_fact__reduce__nat_0) ).

thf(4547,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ c_Fact_Ofact__class_Ofact ) )
        = ( tc_nat @ ( tc_nat @ ( tc_nat @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Fact_Ofact__class_Ofact ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).

thf(270,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__1_0) ).

thf(2436,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(1259,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__numeral__1__right_0) ).

thf(4742,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_Onumber__class_Onumber__of ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1259]) ).

thf(211,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__negative__iff__negative_0) ).

thf(2297,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(1476,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__diff__diff_0) ).

thf(5255,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( c_Suc @ hAPP ) @ ( tc_nat @ ( B @ ( A @ ( c_Suc @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1476]) ).

thf(1559,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_power__gt1__lemma_0) ).

thf(5445,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1559]) ).

thf(349,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_divide__le__eq__number__of1_9) ).

thf(2615,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( A @ ( C @ c_Int_Onumber__class_Onumber__of ) @ ( B @ c_HOL_Oinverse__class_Odivide ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Int_Onumber )
      | ~ ( A @ class_Ring__and__Field_Odivision__by__zero )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(1124,axiom,
    ! [A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_number__of__Pls_0) ).

thf(4423,plain,
    ! [A: $i] :
      ( ( ( A @ ( c_Int_OPls @ c_Int_Onumber__class_Onumber__of ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).

thf(1329,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__1__eq__mult__iff_1) ).

thf(4908,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Oone__class_Oone ) )
      | ( ( tc_nat @ c_HOL_Oone__class_Oone )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1329]) ).

thf(45,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__fieldgb_Oinverse__divide_0) ).

thf(1917,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oinverse__class_Odivide ) ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Ofield ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(654,axiom,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_ordered__idom_Omult__less__cancel__left1_1) ).

thf(3316,plain,
    ! [M: $i,L: $i,K: $i,J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( M @ ( L @ ( G @ ( C @ ( K @ ( A @ ( D @ ( J @ ( F @ ( I @ ( H @ c_Ring__and__Field_Oordered__idom ) ) ) ) ) ) ) ) ) ) )
      | ~ ( B @ ( E @ ( G @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( F @ ( E @ ( D @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(442,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_inverse__negative__imp__negative_0) ).

thf(2834,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Oord__class_Oless ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__field ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(1651,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_less__SucI_0) ).

thf(5664,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1651]) ).

thf(234,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__ringb_Oadd__r0__iff_1) ).

thf(2351,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(1023,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_class__semiring_Omul__c_0) ).

thf(4189,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).

thf(122,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_neg__0__equal__iff__equal_1) ).

thf(2087,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
        = ( A @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(917,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_even__even__mod__4__iff_0) ).

thf(3933,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Parity_Oeven__odd__class_Oeven ) )
      | ( tc_nat @ ( tc_nat @ ( tc_nat @ ( c_Int_OPls @ c_Int_OBit1 @ c_Int_OBit0 @ c_Int_OBit0 @ c_Int_Onumber__class_Onumber__of ) @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Parity_Oeven__odd__class_Oeven ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(754,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_Suc__le__mono_1) ).

thf(3553,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( c_Suc @ hAPP ) @ ( A @ ( c_Suc @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(1180,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_mult__le__cancel1_0) ).

thf(4552,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).

thf(1070,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_n__less__m__mult__n_0) ).

thf(4298,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).

thf(366,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nonzero__inverse__minus__eq_0) ).

thf(2655,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( A @ ( B @ c_HOL_Ouminus__class_Ouminus ) @ c_HOL_Oinverse__class_Oinverse ) )
        = ( A @ ( A @ ( B @ c_HOL_Oinverse__class_Oinverse ) @ c_HOL_Ouminus__class_Ouminus ) ) )
      | ~ ( A @ class_Ring__and__Field_Odivision__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(947,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__power__eq__Suc__0__iff_2) ).

thf(4002,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) @ ( tc_nat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Suc @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(25,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_order__less__imp__le_0) ).

thf(1870,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(1256,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',cls_nat__le__linear_0) ).

thf(4734,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1256]) ).

thf(10823,plain,
    $false,
    inference(e,[status(thm)],[2452,4447,5561,5896,4729,3434,2876,5390,3449,5999,3927,4877,2250,5810,5659,3021,5437,2822,2630,3873,5469,3944,5365,2612,4094,5088,2744,4835,2889,1823,4450,2341,4571,3053,2336,2109,5593,2031,4262,5329,3661,4062,3399,1995,2921,3698,3979,2787,5654,5676,2595,1850,2427,1982,6042,4682,3566,5028,1956,1950,4992,2280,4904,3990,3072,4075,2776,4960,4747,4479,4532,2077,4043,3534,5454,3185,4037,2976,4376,2117,2104,5225,5168,4226,5820,4339,3905,4515,3813,3230,2381,5644,2908,5842,2062,3549,3509,5242,3377,2527,3040,3153,2953,2590,3460,2947,5878,3098,2671,2676,2014,3910,4603,3012,2099,5224,5669,5405,2837,4326,5265,3498,4215,5691,4614,2854,5237,6031,5529,4973,3217,5906,4870,2316,5316,3616,3693,5161,4247,4980,3717,4762,3848,2195,5967,5425,3794,4633,4090,5361,3049,3745,4418,4739,2607,2686,3004,2035,4647,4011,3135,4097,4026,3044,5288,4110,4528,3338,2933,5257,4638,1872,3781,4440,4650,5713,2476,2627,2168,3313,5782,1954,2844,4435,2042,5412,6046,3276,3103,5977,5393,5745,5770,4030,2173,5533,4792,1835,2216,3236,2348,5227,2972,2046,2985,4496,5518,4266,4618,3088,3308,2010,3947,4124,2886,5903,3281,3466,4760,4963,4156,5269,3880,3878,5750,4948,2531,4467,5352,5342,3263,2698,4692,5759,3574,3061,2708,2501,4334,3612,3485,3776,2185,3713,3820,3891,3248,3204,5953,3442,3976,2328,3195,3029,5306,4896,6034,6039,4070,2667,4151,2755,5910,2682,3125,1826,3749,2562,4023,1832,5989,1992,5289,5706,4568,3353,3803,5108,5279,5461,2703,5744,1922,3638,5802,5480,3084,3951,5076,3866,5617,4669,6049,5338,2713,3998,3840,3463,3374,3385,2857,5866,4862,4358,2049,2552,4664,2767,5193,5604,4622,3919,5323,4928,5052,2929,4675,3470,5851,3902,3734,3014,4141,2024,3835,3708,4383,2301,4460,3180,4390,4507,1875,2852,4771,4253,4008,5974,5792,4432,6002,4396,2794,2469,3670,4815,2291,2017,2178,5682,5787,4411,2950,4696,2237,4329,3887,3199,5888,5526,4443,3075,5597,2090,3393,4857,3724,4820,5856,3527,3473,5946,4050,2122,3525,1864,3973,5804,1978,5541,5914,5032,5156,6054,3569,5805,5985,3923,3300,3093,4115,5761,4556,5259,2073,4162,3851,4751,4055,5210,5101,1886,5321,4924,4825,4701,2103,2762,3389,5572,2508,5059,3986,2058,2367,3537,1928,5773,2833,5401,3189,5379,3739,2190,3421,5375,5384,3808,2523,3036,4364,4087,4996,5296,2772,3157,4040,3361,3756,6021,2809,2242,2639,4643,1943,2965,5629,4519,5137,4707,3870,3606,5824,5328,4211,2718,2205,2586,2884,4082,4065,5695,5298,3899,3857,4753,3984,4531,5142,1915,3679,3151,2011,5012,5875,4893,4323,3591,4589,4407,6015,3720,3531,3635,5443,2364,2164,5545,5070,5291,5034,3995,5117,3318,5862,3767,4576,2307,2083,2496,2621,4312,2660,4059,2905,5334,1868,4172,4999,5122,4925,5206,5049,1900,4712,3827,2541,5794,5579,5719,5085,5754,5893,4160,4915,2409,5611,5734,2560,2805,4989,5044,5349,2147,5266,5956,5815,3742,4344,4128,3239,3233,3884,2132,2414,3250,2396,3514,3682,3778,3889,3471,5924,3286,1883,5983,4192,3718,2211,2653,2958,3948,3166,2445,4295,3191,1938,4280,5879,5702,5302,5492,5937,3269,4608,1920,4053,2433,4725,1968,3090,2565,2826,2820,4947,3137,3065,5177,4994,3297,2872,5021,5988,3894,2068,2200,2609,4187,2179,5551,3546,5038,4402,4942,4979,5201,5566,2327,3028,2345,2547,2313,3763,2492,4104,3727,2741,4732,2228,3647,2112,4080,5993,3931,1910,4680,3799,2769,3445,4316,5285,5189,4957,5673,5423,5635,3963,5687,3968,5313,2245,2498,3615,3831,2840,2115,5145,2515,3254,2100,3413,1851,5098,3122,2377,2801,5345,4248,5615,6028,2359,4486,5502,5798,1970,4048,4544,4429,4636,2692,4525,4845,5428,4233,1878,2726,2724,4930,3632,3119,4840,4883,5847,3650,3916,3941,4219,2268,3440,5497,3001,5951,3176,3018,4817,4898,4676,3143,3622,4167,2419,4513,3428,2085,2460,2798,3954,4049,5941,4653,5830,3141,4951,3311,3326,5166,5818,1964,4276,4582,2323,4983,3846,3407,5651,3210,5620,3596,4421,3010,2080,5253,2196,2240,1842,5547,5213,3032,1937,3643,4867,5905,1889,3555,4457,2978,2649,3294,5387,4736,2781,4530,3935,5273,2125,4384,3937,1947,1984,6005,5369,2140,5245,4804,3242,3703,2285,5576,2968,5063,2536,3541,2487,3059,2387,2153,2900,5639,5134,3770,1974,1932,4244,2581,2465,4934,4540,2006,4231,2813,4347,3409,2862,2052,4902,3333,4953,3109,4689,3920,4416,3364,3201,4223,3476,4535,3225,4782,5415,6011,2402,4906,2221,1895,2529,5170,3147,3454,2634,1857,5557,4184,5112,2749,5310,5153,4352,2203,5711,3563,2761,4195,3624,4625,5090,3587,3183,4966,4765,3173,4476,3306,1846,5434,5964,2963,5447,2645,5513,5198,3841,2617,5126,5840,2662,4004,2338,3675,4308,2909,4550,2144,2001,5666,4685,2391,3042,3491,5230,4393,2894,3343,4921,1906,5679,2729,4919,3130,5722,4131,5808,1959,2556,4354,2253,2033,2149,4594,5969,3257,5451,4874,5508,2570,2941,4562,3262,5608,2602,2028,2734,5928,5016,5882,5095,4269,5647,4017,5715,4489,6052,3396,2376,3070,2970,2489,5962,4465,3458,4328,2695,5698,4144,3372,4000,3387,4662,5456,3925,2859,2938,5474,2225,3504,4283,5488,4494,3811,5395,3240,2789,3519,5844,3609,2849,2525,5065,2188,2981,2393,3086,3736,3758,5627,3266,2674,2568,5612,4032,2806,2991,3604,4381,5271,3404,3868,4758,4769,4505,3599,5965,3023,3773,3751,5363,3340,5148,4273,3302,3223,2161,4796,2891,4837,3077,4645,3646,4112,5859,3695,4099,3330,2738,4149,3208,5282,4236,3663,2935,4822,2442,2759,3783,3487,2055,2706,5449,2166,3619,3579,2457,4854,2425,3416,5997,3304,5784,2474,4887,5432,2573,2319,2510,4716,6024,2721,2538,3419,3732,2299,4586,3710,5972,4107,2065,3101,2521,4790,4337,2679,5671,4498,4369,2151,2642,3155,3815,2214,2624,5103,5520,3551,2422,3038,2637,3957,2774,3287,4828,5303,5827,2097,5853,3270,4601,3106,3468,5054,2669,5222,4213,4616,3197,5346,4073,5531,2856,4176,5689,4300,4374,3066,2896,3456,3747,2632,3133,5082,5175,2757,4181,3006,5466,3861,2578,5552,2824,4203,5114,2182,5367,4430,5584,5920,4420,4445,2605,5730,3668,4462,4911,2261,1821,4264,3572,4452,5599,3034,1838,5656,5318,2874,5885,3483,3583,6044,2647,2289,5180,2278,5007,2974,5410,2842,5838,1952,5780,4554,3480,5742,5931,5482,4502,5516,2931,3016,5849,4703,2967,2303,3908,2867,3539,2075,4933,5057,2439,3627,5372,2171,5216,3838,2765,2550,2481,3687,5248,3806,3159,2369,4362,2592,2752,3706,5725,6027,3178,5403,2999,5693,3348,2156,5495,4067,2207,5326,4890,4481,4158,2506,3380,4084,5959,3193,2107,3161,4319,4879,2927,5807,6008,4812,4986,5812,5661,5463,3522,4331,2054,4426,4207,4405,2914,2816,5538,5587,4136,2096,2665,4458,3576,4260,2994,5025,2784,2271,3206,2471,4598,5757,2037,3981,2878,2503,4292,2449,1990,4859,2684,4688,2333,2846,3278,2597,5358,4722,5570,5822,2715,3640,3913,3112,3214,3904,5748,3833,4469,4971,3988,2513,1845,4749,4255,3391,5624,5229,1961,4035,4360,5294,5927,5602,4937,5871,2060,4939,3082,2192,4673,5499,4950,3260,2588,2732,4200,5980,5523,2518,4517,4641,5721,2175,2407,3786,5079,4630,2831,2689,3080,5431,5912,5753,5555,3823,5106,3769,4409,3246,4437,4522,4314,5262,5235,5767,4092,4240,2092,1987,5420,6036,5418,2454,5731,3818,3114,3273,3791,3063,2920,3722,4020,3655,5948,2276,2219,1980,5203,5030,3801,5309,5995,3127,5001,2701,4241,2545,2599,3359,3329,2044,5186,5277,3715,2354,2247,2379,4202,4695,4342,5717,4414,4217,5208,1934,3697,2411,3488,4802,5902,4057,1898,1849,3652,2558,3970,3829,5736,4133,2992,4611,4976,2543,3356,5240,4510,2357,1972,2711,4046,4538,3117,2860,3961,5472,1881,3914,5100,4795,5596,3366,3007,2266,2584,4257,4671,4819,3684,3288,3589,1966,3447,2325,2209,2865,5832,3168,1940,2435,4639,3284,2013,2924,3478,5939,4289,4278,3139,5873,5490,1839,3584,3452,2458,4138,4698,2479,4851,4606,3401,4727,5917,4945,2818,3067,2870,4165,2828,5536,6013,3171,5777,4944,4873,2078,1913,4656,2310,5573,3188,5510,4559,5934,5421,4078,5907,2997,4102,3346,4006,3320,2305,5124,4170,2342,3056,5485,5332,1866,2071,2658,3929,5068,2907,4310,2792,3665,4678,1998,4574,3797,3593,4913,2273,3789,3825,3383,4250,1854,4881,5922,5632,2443,3729,4121,4710,2988,3864,2283,2389,4806,1903,4666,2135,5382,2554,2532,5191,5800,5132,3336,5899,3369,4126,3220,4542,4715,2004,2026,3630,3501,1822,4834,3516,4495,4705,2130,3859,3252,5505,4900,2262,1861,2384,4197,5408,4153,6018,5739,5864,4745,2447,2944,5963,2956,2796,5164,2912,4321,2399,3430,4591,4774,3598,2462,5023,3149,2330,3896,2198,4387,2576,3677,2362,5549,5568,3351,5869,3993,2902,2120,5119,2095,5590,2230,1893,2294,2494,5196,2917,4372,3761,4306,5704,5336,2779,2019,4619,2374,2287,4917,5009,5728,5004,4596,4719,5764,5477,4455,1908,2142,3379,3658,2258,5051,4014,5129,4472,4399,3511,5606,4378,2534,2255,3291,4356,5796,4809,2983,3601,2233,3882,5954,5232,5356,2604,4843,1876,2736,4787,2008,1986,4491,3496,4579,4885,5014,3690,5398,4367,5041,3164,5700,3956,5300,3850,2961,2223,2040,5559,3145,1859,4146,4029,4178,4303,4565,4777,5159,4350,4755,5092,3560,5709,4286,3558,5990,4210,3875,4474,4968,2746,5019,5835,5564,2417,2366,5339,2404,3323,3426,5251,1891,5036,4865,5378,3765,5642,2619,3424,3506,3843,5200,3673,2021,5110,5440,2651,3244,1829,5151,4484,3965,3886,4955,4501,4271,2138,1930,5275,2127,4848,2159,2810,3025,1827,2430,3543,2803,4799,2898,3411,5140,5637,2265,4118,5581,2467,5061,1976,5975,4239,4831,3392,5623,4229,5046,4206,5774,4659,1925,5219,3939,2539,5891,5354,2484,3494,4221,2372,2879,3580,5073,4767,1848,3432,3052,3047,4174,3753,3960,3528,5172,4826,5943,3437,4779,2321,3227,5542,4627,4584,1962,2881,5986,1945,3212,5649,5458,5789,5618,4784,3096,3854,2690,5183,5261,4547,2436,4742,2297,5255,5445,2615,4423,4908,1917,3316,2834,5664,2351,4189,2087,3933,3553,4552,4298,2655,4002,1870,4734]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02  % Problem  : SWV706-1 : TPTP v9.3.1. Released v4.1.0.
% 0.00/0.06  % Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.12/10.64  % Computer : n007.cluster.edu
% 0.12/10.64  % Model    : x86_64 x86_64
% 0.12/10.64  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/10.64  % Memory   : 8046.5625MB
% 0.12/10.64  % OS       : Linux 6.8.0-71-generic
% 0.12/10.64  % CPULimit : 300
% 0.12/10.64  % WCLimit  : 300
% 0.12/10.64  % DateTime : Sat Sep 26 14:46:35 UTC 2026
% 0.12/10.65  % CPUTime  : 
% 0.12/10.65  Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.92/11.61  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 2.52/12.39  % [INFO] 	 Parsing done (778ms). 
% 2.72/12.41  % [INFO] 	 Running in sequential loop mode. 
% 3.14/12.80  % [INFO] 	 eprover registered as external prover. 
% 3.14/12.81  % [INFO] 	 Scanning for conjecture ... 
% 4.07/13.29  % [INFO] 	 Found a conjecture (or negated_conjecture) and 1824 axioms. Running axiom selection ... 
% 5.97/13.98  % [INFO] 	 Axiom selection finished. Selected 1817 axioms (removed 7 axioms). 
% 8.09/14.80  % [INFO] 	 Problem is propositional (TPTP CNF). 
% 8.09/14.87  % [INFO] 	 Type checking passed. 
% 8.09/14.88  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 82.85/34.68  % External prover 'e' found a proof!
% 82.85/34.69  % [INFO] 	 Killing All external provers ... 
% 82.85/34.69  % Time passed: 23829ms (effective reasoning time: 22270ms)
% 82.58/34.72  % Axioms used in derivation (1817): cls_abs__le__zero__iff_0, cls_mult__eq__1__iff_1, cls_Suc__pred_0, cls_divide__le__eq__number__of_9, cls_Suc__Suc__mult__two__diff__two_0, cls_root__cancel_0, cls_mod__Suc__eq__mod__add3_0, cls_add__0__left_0, cls_le__divide__eq_8, cls_add__cancel__end_1, cls_minus__le__iff_0, cls_even__sum__nat_2, clsarity_nat__OrderedGroup_Oab__semigroup__add, cls_less__divide__eq_3, cls_divide__le__eq__number__of1_9, cls_zero__less__power__eq_1, cls_not__one__less__zero_0, cls_ordered__idom_Omult__less__cancel__left2_4, cls_mult__less__cancel__left__disj_5, cls_less__iff__diff__less__0_0, cls_fact__mono__nat_0, cls_lemma__even__mod__4__div__2_0, cls_ordered__idom_Oodd__0__le__power__imp__0__le_0, cls_even__add_2, cls_zero__le__power__iff_2, cls_add__Suc__shift_0, cls_xt1_I9_J_0, cls_linorder__antisym__conv2_1, cls_one__le__mult__iff_1, cls_add__pos__nonneg_0, cls_even__difference__nat_4, cls_gb__semiring_Osemiring__rules_I33_J_0, cls_power__0_0, cls_mult__numeral__1__right_0, cls_ordered__idom_Ozero__le__even__power_H_0, cls_lemma__not__even__div2_0, cls_le__0__eq_1, cls_natgb_Oadd__r0__iff_1, cls_divide__eq__eq_3, cls_eq__add__iff2_1, cls_ordered__idom_Omult__le__cancel__left1_5, clsarity_Complex__Ocomplex__Ring__and__Field_Oring__1, cls_mult__div__cancel_0, cls_sum__power2__gt__zero__iff_2, cls_neg__0__le__iff__le_1, cls_comm__monoid__add_Opower__mult__distrib_0, cls_le__divide__eq_3, cls_Rational_Oordered__idom_Osgn__greater_1, cls_order__le__less__trans_0, cls_mult__cancel2_0, cls_add__is__1_1, cls_less__diff__iff_1, cls_mult__zero__right_0, cls_zero__less__one_0, cls_odd__power__less__zero_0, cls_Suc__inject_0, cls_eq__imp__le_0, cls_Suc__mono_0, cls_nat__0__less__mult__iff_1, cls_le__diff__iff_0, cls_zero__less__divide__iff_5, cls_monoid__mult_Opower__add_0, cls_gb__semiring_Osemiring__rules_I12_J_0, cls_even__num__iff_1, cls_mod__mult__mult1_0, cls_less__divide__eq__number__of1_6, cls_abs__div__pos_0, clsarity_Int__Oint__OrderedGroup_Opordered__ab__group__add__abs, cls_zero__le__odd__power_1, cls_abs__if__lattice_0, cls_abs__le__iff_2, cls_less__int__code_I15_J_1, cls_odd__add_1, cls_divide__le__eq_2, cls_ordered__idom_Ozero__eq__power2_0, cls_mult__imp__le__div__pos_0, cls_fact__add__num__eq__if__nat_1, cls_divide__pos__pos_0, clsarity_Int__Oint__Int_Onumber, cls_class__ringb_Oadd__r0__iff_1, cls_mult__nonneg__nonneg_0, cls_mult__2__right_0, cls_power2__eq__imp__eq_0, cls_eq__divide__eq__number__of1_4, cls_add__less__cancel__left_1, cls_mult__right__mono_0, cls_mult__right_Ozero_0, cls_monoid__mult_Opower__commutes_0, cls_nat__mult__eq__1__iff_1, cls_gb__semiring_Omul__0_0, cls_DERIV__mult__lemma_0, cls_order__less__le__trans_0, cls_rel__simps_I12_J_0, cls_order__eq__iff_0, cls_divide__less__0__iff_0, cls_div__le__mono_0, cls_comm__semiring__class_Odistrib_0, cls_mod2__gr__0_0, cls_not__iszero__1_0, clsarity_Complex__Ocomplex__RealVector_Oreal__normed__algebra, cls_gb__semiring_Osemiring__rules_I7_J_0, cls_even__zero__nat_0, cls_even__difference__nat_1, cls_divide__eq__eq__number__of_4, cls_le__add__diff__inverse2_0, cls_abs__of__nonneg_0, cls_Nat_Odiff__diff__eq_0, cls_zero__le__even__power_H_0, cls_le__diff__conv_1, cls_le__minus__self__iff_0, cls_left__add__mult__distrib_0, cls_mult__less__cancel__right__disj_0, cls_divide__self_0, cls_rel__simps_I43_J_1, cls_mult__nonneg__nonpos2_0, cls_power__divides__special_0, cls_one__is__add_4, clsarity_Complex__Ocomplex__Ring__and__Field_Ofield, cls_square__eq__iff_1, cls_le__divide__eq__number__of1_0, cls_minus__zero_0, cls_power__le__zero__eq__number__of_1, cls_diff__Suc__1_0, cls_nat__number__of__add__1_0, cls_Lim_Ominus__diff__minus_0, cls_mult_Odiff__left_0, clsarity_Int__Oint__Ring__and__Field_Olordered__ring, cls_inverse__zero__imp__zero_0, cls_xt1_I12_J_0, cls_less__int__code_I16_J_0, cls_sum__squares__eq__zero__iff_0, cls_negative__imp__inverse__negative_0, cls_less__special_I3_J_0, cls_even__even__mod__4__iff_1, clsarity_Int__Oint__Ring__and__Field_Oabs__if, cls_Suc__n__not__n_0, cls_Bit0__Pls_0, cls_fact__add__num__eq__if2__nat_0, cls_left__minus_0, cls_ordered__idom_Omult__le__cancel__right1_2, cls_less__special_I1_J_1, cls_nat__1__eq__mult__iff_0, cls_divide__le__eq__number__of_4, cls_zero__less__power__eq_5, cls_ordered__idom_Onot__sum__power2__lt__zero_0, clsarity_Complex__Ocomplex__Int_Onumber, cls_divide__nonpos__neg_0, cls_class__semiring_Osemiring__rules_I9_J_0, cls_mult__less__cancel__left__disj_1, cls_zero__le__one_0, cls_nat__mult__div__cancel__disj_0, cls_comp__arith_I112_J_0, cls_number__of__Bit1_0, cls_mult__is__0_2, cls_ordered__idom_Oabs__sgn_0, cls_nat__add__left__cancel__less_1, cls_power__eq__0__iff_1, cls_ordered__idom_Omult__less__cancel__right1_5, cls_termination__basic__simps_I5_J_0, clsarity_Complex__Ocomplex__OrderedGroup_Ocomm__monoid__mult, cls_not__iszero__Numeral1_0, cls_eq__diff__iff_0, cls_divide__numeral__1_0, cls_less__antisym_0, cls_nat__mod__eq__iff_1, cls_divide__less__eq__number__of1_5, cls_le__divide__eq__number__of_6, cls_not__sum__squares__lt__zero_0, cls_diff__add__cancel_0, cls_pos__add__strict_0, cls_half_0, cls_nat__lt__two__imp__zero__or__one_0, cls_ordered__idom_Osum__power2__le__zero__iff_0, cls_add__increasing2_0, cls_gb__semiring_Opwr__pwr_0, cls_iszero__number__of__Bit0_0, cls_even__num__iff_0, cls_inverse__eq__iff__eq_0, cls_less__half__sum_0, cls_ordered__idom_Oabs__power__minus_0, cls_ordered__idom_Opower2__abs_0, cls_abs__diff__triangle__ineq_0, cls_ordered__idom_Omult__le__cancel__left1_0, cls_mod__by__0_0, cls_add__is__1_2, cls_power__inject__exp_0, cls_ordered__idom_Oabs__mult__self_0, cls_zero__le__mult__iff_3, cls_class__semiring_Osemiring__rules_I29_J_0, cls_abs__triangle__ineq2_0, cls_linorder__neqE_0, cls_divide__1_0, cls_add__strict__right__mono_0, clsarity_Int__Oint__Orderings_Oorder, cls_one__le__mult__iff_0, cls_even__Suc_0, cls_power__increasing_0, cls_power2__less__0_0, cls_minus__one__even__power_0, cls_inverse__le__1__iff_2, cls_less__add__Suc2_0, clsarity_Int__Oint__OrderedGroup_Olordered__ab__group__add, cls_divide__le__eq__number__of1_4, clsarity_Int__Oint__Ring__and__Field_Opordered__ring, cls_less__divide__eq__number__of_8, cls_nonzero__inverse__eq__divide_0, cls_nat__1__add__1_0, clsarity_Int__Oint__OrderedGroup_Omonoid__add, cls_right__distrib__number__of_0, cls_lemma__realpow__diff_0, cls_power3__eq__cube_0, clsarity_Int__Oint__Ring__and__Field_Osemiring, cls_class__semiring_Osemiring__rules_I20_J_0, cls_zero__le__divide__iff_2, cls_abs__minus__le__zero_0, cls_power__le__zero__eq_3, cls_half__gt__zero_0, cls_class__semiring_Osemiring__rules_I21_J_0, cls_divide__less__eq__number__of1_0, cls_divide__less__eq__number__of_1, cls_less__imp__inverse__less__neg_0, cls_class__semiring_Osemiring__rules_I10_J_0, cls_zero__less__power__eq_4, cls_divide__le__eq_7, cls_zero__less__power__nat__eq__number__of_0, cls_frac__le_0, cls_left__distrib__number__of_0, cls_power2__diff_0, cls_div__add__self2_0, cls_mod__add__eq_0, cls_number__of__Pls_0, cls_termination__basic__simps_I1_J_0, cls_divide__less__eq_4, cls_minus__add__distrib_0, cls_zero__less__divide__iff_0, cls_abs__eq__mult_1, cls_split__div__lemma_1, cls_add__le__cancel__left_1, cls_divide__le__eq_8, cls_less__divide__eq__number__of_3, clsarity_Complex__Ocomplex__Ring__and__Field_Osemiring__1, cls_rel__simps_I4_J_1, cls_gb__semiring_Opwr__Suc_0, cls_power__le__zero__eq__number__of_2, cls_add__is__0_0, clsarity_nat__OrderedGroup_Omonoid__add, cls_le__divide__eq__number__of1_5, cls_Suc__lessI_0, cls_diff__cancel_0, cls_add__imp__eq_0, cls_gb__semiring_Osemiring__rules_I3_J_0, cls_abs__mult__less_0, cls_divide__le__0__1__iff_1, cls_rel__simps_I8_J_0, cls_nat__eq__add__iff1_0, cls_mod__add__left__eq_0, cls_split__mod_1, cls_less__Suc__eq_2, cls_Suc__mult__two__diff__one_0, cls_odd__nat__equiv__def_1, cls_not__less__eq_1, cls_divide__Numeral0_0, cls_nat__add__left__cancel__less_0, cls_minus__diff__eq_0, cls_mult__nonpos__nonneg_0, cls_power__less__zero__eq_1, cls_one__is__add_1, cls_diff__nat__number__of_0, cls_divide__Numeral1_0, cls_inverse__less__1__iff_2, cls_add__le__cancel__right_1, cls_gb__semiring_Osemiring__rules_I28_J_0, cls_one__le__inverse__iff_2, cls_less__nat__number__of_0, cls_inverse__1_0, cls_minus__le__self__iff_0, cls_minus__divide__right_0, cls_inverse__negative__iff__negative_1, cls_less__0__number__of_0, clsarity_nat__Orderings_Olinorder, cls_eq__divide__eq__number__of_4, cls_power__le__imp__le__exp_0, cls_mult__less__imp__less__right_0, cls_comm__monoid__add_Omult__assoc_0, cls_order__less__trans_0, cls_class__semiring_Osemiring__rules_I14_J_0, cls_power__0__left__number__of_0, cls_add__cancel__21_1, cls_mod__div__equality_0, cls_left__diff__distrib__number__of_0, cls_divide__less__eq__number__of1_6, cls_div__mult__self1__is__m_0, cls_nat__power__eq__Suc__0__iff_1, cls_divide__eq__eq_0, cls_mod__mult__self1__is__0_0, cls_mult__nonpos__nonpos_0, cls_class__ring_Osub__add_0, cls_class__ringb_Onoteq__reduce_0, cls_rel__simps_I49_J_0, cls_mod__mult__right__eq_0, cls_comm__monoid__add_Omult__1__right_0, cls_nat__eq__add__iff2_0, cls_split__div_H_6, cls_add__leE_0, cls_minus__minus_0, cls_ordered__idom_Omult__less__cancel__right1_4, cls_sum__squares__ge__zero_0, cls_power__one_0, cls_add__diff__assoc2_0, cls_power__gt1_0, cls_add__nonpos__neg_0, clsarity_Complex__Ocomplex__OrderedGroup_Ogroup__add, cls_zero__less__power__eq__number__of_1, cls_Suc__mult__cancel1_0, cls_le__eq__neg_0, cls_left__inverse_0, cls_abs__of__pos_0, cls_divide__less__eq_5, cls_ordered__idom_Omult__le__cancel__left1_3, cls_neg__less__iff__less_1, cls_nat__less__add__iff1_0, cls_mod__diff__eq_0, cls_Suc3__eq__add__3_0, cls_eq__divide__eq__number__of_1, cls_zero__eq__power2_0, cls_zero__le__power__eq_2, cls_nat__mult__1_0, cls_ordered__idom_Omult__less__cancel__left2_0, cls_ordered__idom_Omult__le__cancel__left2_0, clsarity_nat__Ring__and__Field_Oordered__comm__semiring__strict, cls_add__left__cancel_0, cls_less__minus__iff_0, cls_ordered__idom_Omult__less__cancel__right2_5, cls_power__less__imp__less__exp_0, cls_mult__le__cancel__left__pos_0, cls_gb__semiring_Osemiring__rules_I13_J_0, cls_natgb_Oadd__mul__solve_0, cls_split__mult__neg__le_0, cls_mult__less__imp__less__left_0, cls_root4_0, cls_mult__minus1_0, cls_split__div_5, cls_inverse__divide_0, clsarity_Int__Oint__Ring__and__Field_Oring__no__zero__divisors, cls_mult__less__cancel1_0, cls_rel__simps_I13_J_0, cls_minus__unique_0, cls_div__by__0_0, cls_inverse__nonnegative__iff__nonnegative_0, cls_nat__less__add__iff2_1, cls_COMBB__def_0, cls_mult__less__cancel__left__disj_4, cls_class__ringb_Oadd__mul__solve_1, cls_le__0__eq_0, cls_divide__le__0__iff_3, cls_sum__squares__le__zero__iff_1, cls_le__divide__eq__number__of1_9, cls_odd__add_0, cls_divide__le__eq__number__of_5, cls_nat__diff__add__eq1_0, cls_ordered__idom_Omult__le__cancel__left2_4, cls_gr0I_0, cls_even__power__nat_1, cls_less__divide__eq_4, cls_zless__le_1, cls_mod__add__self2_0, cls_mult__less__cancel2_1, cls_add__gr__0_0, cls_mult__less__cancel__right__disj_3, cls_ordered__idom_Omult__le__cancel__right2_0, cls_eq__add__iff1_0, cls_ordered__idom_Osum__power2__eq__zero__iff_0, clsarity_Int__Oint__Ring__and__Field_Oring__1, cls_mult__Suc__right_0, cls_add__2__eq__Suc_H_0, cls_power__less__zero__eq__number__of_1, cls_mult__eq__0__iff_0, cls_add__is__0_1, clsarity_nat__Ring__and__Field_Oordered__semidom, cls_zero__power2_0, cls_Suc__double__inj__on_0, cls_add__nonneg__pos_0, cls_less__special_I3_J_1, cls_even__product__nat_1, cls_ordered__idom_Osgn__0__0_1, cls_sum__power2__le__zero__iff_2, clsarity_Int__Oint__OrderedGroup_Ocancel__semigroup__add, cls_divide__strict__left__mono_0, cls_gb__semiring_Oadd__c_0, cls_abs__less__iff_2, clsarity_nat__OrderedGroup_Opordered__comm__monoid__add, clsarity_Int__Oint__Ring__and__Field_Ocomm__semiring__1, cls_divide__less__eq__number__of_6, clsarity_nat__OrderedGroup_Omonoid__mult, cls_gr__implies__not0_0, cls_diff__diff__cancel_0, cls_mult__left__mono__neg_0, cls_less__divide__eq__number__of1_2, cls_n__less__n__mult__m_0, cls_nat__less__cases_0, cls_eq__divide__eq_4, cls_mult__mono_0, cls_diff__diff__right_0, clsarity_Int__Oint__OrderedGroup_Olordered__ab__group__add__abs, clsarity_nat__Ring__and__Field_Osemiring__0, cls_abs__power__minus_0, cls_nat__mult__eq__cancel1_0, cls_diff__diff__left_0, cls_ab__left__minus_0, cls_mod__by__1_0, clsarity_Complex__Ocomplex__OrderedGroup_Omonoid__mult, cls_zero__less__double__add__iff__zero__less__single__add_0, cls_class__semiring_Omul__0_0, cls_right__inverse__eq_0, cls_rel__simps_I38_J_1, cls_field__power__not__zero_0, clsarity_nat__Ring__and__Field_Oordered__semiring, cls_neg__equal__iff__equal_0, cls_frac__eq__eq_0, cls_power__m1__odd_0, clsarity_nat__Divides_Osemiring__div, cls_one__le__power_0, cls_diff__le__mono2_0, cls_divide__eq__eq_4, cls_Suc__le__mono_0, cls_one__add__one__is__two_0, cls_fact__nonzero__nat_0, cls_add__le__mono_0, cls_order__less__asym_0, cls_even__power__le__0__imp__0_0, cls_class__semiring_Osemiring__rules_I7_J_0, cls_abs__add__abs_0, clsarity_Int__Oint__Ring__and__Field_Oordered__semiring__strict, cls_power_Opower_Opower__0_0, cls_linorder__not__less_0, cls_double__add__less__zero__iff__single__less__zero_0, cls_power__le__zero__eq__number__of_4, cls_power__even__abs_0, cls_monoid__mult_Opower__Suc2_0, cls_le__antisym_0, cls_add__right__mono_0, cls_rel__simps_I17_J_0, cls_nat__number__of__Pls_0, cls_add__less__mono_0, cls_rel__simps_I2_J_0, cls_gb__semiring_Omul__c_0, cls_less__add__iff2_0, clsarity_nat__Ring__and__Field_Ocomm__semiring, cls_mult_Ozero__left_0, cls_divide__neg__neg_0, cls_mult__2_0, cls_mod__if_1, cls_less__iff__diff__less__0_1, cls_inverse__le__1__iff_1, cls_power__eq__0__iff__number__of_2, cls_less__divide__eq__number__of1_0, cls_power__0__left_0, cls_double__eq__0__iff_1, cls_eq__number__of_0, cls_mult__strict__right__mono_0, cls_class__semiring_Omul__c_0, cls_rel__simps_I12_J_1, cls_power__minus1__even_0, cls_mod__minus__eq_0, cls_mult__pos__neg_0, cls_zero__less__power__eq__number__of_4, cls_nat__0__less__mult__iff_0, cls_div__mult__self2__is__id_0, cls_order__eq__refl_0, cls_zero__le__mult__iff_0, cls_le__divide__eq__number__of_7, cls_divide__eq__eq__number__of1_0, cls_inverse__eq__1__iff_1, cls_mult__strict__mono_0, clsarity_Int__Oint__OrderedGroup_Opordered__comm__monoid__add, clsarity_Complex__Ocomplex__Ring__and__Field_Omult__zero, cls_iszero__number__of__Pls_0, clsarity_Int__Oint__Ring__and__Field_Omult__mono, cls_abs__minus__one_0, cls_div__mult2__eq_0, cls_order__le__less_0, cls_order__less__asym_H_0, cls_monoid__mult_Omult__1__left_0, cls_rel__simps_I13_J_1, cls_Rational_Oordered__idom_Osgn__less_0, cls_mult_Ozero__right_0, cls_inverse__minus__eq_0, 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cls_diff__frac__eq_0, cls_power__eq__0__iff_0, cls_eq__eqI_1, cls_Suc__n__not__le__n_0, cls_class__semiring_Oadd__c_0, cls_zero__less__diff_1, cls_div__add1__eq_0, cls_nonzero__power__divide_0, cls_one__less__power_0, cls_divide__le__eq__number__of1_8, clsarity_Int__Oint__OrderedGroup_Oab__group__add, cls_mult__pos__pos_0, cls_less__int__code_I13_J_1, cls_less__eqI_0, cls_nat__add__left__cancel_0, cls_add__le__mono1_0, cls_neg__0__equal__iff__equal_0, clsarity_Int__Oint__Ring__and__Field_Omult__mono1, cls_mult__strict__right__mono__neg_0, clsarity_nat__Ring__and__Field_Ono__zero__divisors, cls_power__diff__inverse_0, cls_add__is__1_3, cls_class__semiring_Osemiring__rules_I36_J_0, cls_divide__less__eq__number__of1_4, cls_div__if_0, cls_add__eq__self__zero_0, cls_inverse__less__imp__less__neg_0, cls_add__mono_0, cls_split__div_H_7, cls_mod__less__divisor_0, cls_rel__simps_I9_J_0, cls_inverse__zero_0, clsarity_Int__Oint__Divides_Osemiring__div, cls_Suc__leD_0, cls_le__divide__eq__number__of1_1, cls_le__diff__iff_1, cls_zero__less__divide__iff_4, cls_add__pos__pos_0, cls_inverse__eq__divide_0, cls_less__0__number__of_1, cls_power__less__imp__less__base_0, cls_split__mod_5, cls_Suc__eq__plus1_0, cls_divide__le__0__iff_0, clsarity_Int__Oint__OrderedGroup_Opordered__ab__semigroup__add, cls_n__not__Suc__n_0, cls_divide_Oadd_0, cls_mult__right__less__imp__less_0, cls_nat__less__add__iff1_1, cls_ordered__idom_Oabs__power2_0, cls_div__2__gt__zero_0, cls_less__minus__self__iff_0, cls_Suc__mod__eq__add3__mod_0, cls_DERIV__inverse__lemma_0, cls_power__eq__if_0, cls_sum__squares__gt__zero__iff_0, cls_ordered__idom_Osum__power2__gt__zero__iff_2, cls_power__less__zero__eq__number__of_0, cls_power__inject__base_0, cls_nat__le__add__iff1_1, cls_less__divide__eq__number__of_2, cls_inverse__le__imp__le_0, cls_equation__minus__iff_1, cls_ordered__idom_Omult__le__cancel__left1_1, cls_fact__less__mono__nat_0, clsarity_Complex__Ocomplex__Ring__and__Field_Osemiring, cls_gb__semiring_Osemiring__rules_I24_J_0, cls_mult_Oprod__diff__prod_0, cls_zmod__simps_I1_J_0, cls_class__semiring_Osemiring__rules_I8_J_0, cls_fact__add__num__eq__if2__nat_1, cls_less__SucI_0, cls_nat__mult__le__cancel1_0, cls_div2__Suc__Suc_0, cls_divide__less__eq_9, cls_add__neg__nonpos_0, cls_eq__divide__eq__number__of1_0, cls_ordered__idom_Osgn__sgn_0, cls_mult__divide__mult__cancel__right_0, cls_diff__add__inverse_0, cls_power__le__zero__eq_4, cls_abs__leI_0, cls_rel__simps_I47_J_0, cls_class__ringb_Osubr0__iff_1, cls_divide__strict__right__mono_0, cls_ordered__idom_Omult__le__cancel__right1_1, clsarity_Complex__Ocomplex__Power_Opower, cls_zero__le__power__eq_1, cls_nat__0__less__mult__iff_2, cls_zero__le__power__eq__number__of_0, cls_odd__Suc__mult__two__ex_0, cls_not__less0_0, cls_inj__Suc_0, cls_mult__less__cancel__left__disj_2, cls_minus__add_0, cls_power__eq__0__iff__number__of_0, cls_nat__mult__assoc_0, cls_div__Suc_0, cls_diffs0__imp__equal_0, cls_frac__less_0, clsarity_Int__Oint__OrderedGroup_Ocomm__monoid__add, cls_expand__Suc_0, cls_iszero__number__of__Bit0_1, cls_ordered__idom_Osgn__1__neg_0, cls_mod__div__decomp_0, clsarity_Int__Oint__OrderedGroup_Opordered__cancel__ab__semigroup__add, cls_divide__inverse_0, cls_field__inverse_0, cls_gb__semiring_Osemiring__rules_I19_J_0, cls_ordered__idom_Omult__le__cancel__left2_5, cls_class__semiring_Opwr__mul_0, cls_abs__le__zero__iff_1, cls_odd__1__nat_0, cls_class__semiring_Osemiring__rules_I24_J_0, cls_abs__eq__0_1, cls_nat__mult__commute_0, cls_mult__less__cancel__right__disj_1, cls_fact__Suc__0__nat_0, cls_less__int__code_I16_J_1, cls_divide__le__eq_9, cls_eq__iff__diff__eq__0_1, cls_even__difference__nat_2, cls_add__less__cancel__left_0, cls_zmod__simps_I4_J_0, cls_odd__nat__div__two__times__two__plus__one_0, cls_abs__one_0, cls_gb__semiring_Omul__a_0, cls_abs__inverse_0, cls_right__minus_0, cls_one__is__add_3, cls_nonzero__minus__divide__divide_0, cls_fact__gt__zero__nat_0, cls_divide__less__eq_6, cls_rel__simps_I40_J_0, cls_ordered__idom_Omult__le__cancel__left1_2, cls_less__divide__eq__number__of1_9, cls_Suc__less__eq_0, cls_even__sum__nat_3, cls_inverse__less__imp__less_0, cls_divide__less__eq__number__of_2, cls_inverse__positive__imp__positive_0, cls_even__mult__two__ex_1, cls_rel__simps_I14_J_1, cls_class__semiring_Osemiring__rules_I25_J_0, cls_divide__eq__eq__number__of_2, cls_mult__eq__self__implies__10_0, cls_gb__semiring_Opwr__0_0, cls_inverse__mult__distrib_0, cls_eq__divide__eq__number__of_0, cls_le__eqI_1, cls_power2__sum_0, cls_power__minus_0, cls_class__semiring_Opwr__0_0, cls_Suc__diff__le_0, cls_mult__eq__0__iff_1, cls_odd__nat__plus__one__div__two_0, cls_le__iff__diff__le__0_1, cls_Bit1__Min_0, cls_gb__semiring_Osemiring__rules_I4_J_0, cls_divide__nonpos__pos_0, cls_add__gr__0_2, cls_diff__le__mono_0, cls_add__less__le__mono_0, cls_zero__le__power_0, cls_power__minus1__odd_0, cls_gb__semiring_Osemiring__rules_I20_J_0, cls_power__increasing__iff_0, cls_divide__less__eq_3, clsarity_nat__Ring__and__Field_Osemiring__1, cls_even__product__nat_0, cls_nonzero__inverse__mult__distrib_0, clsarity_Complex__Ocomplex__Ring__and__Field_Ocomm__semiring__1, cls_nonzero__power__inverse_0, cls_less__divide__eq_0, cls_nat__power__eq__Suc__0__iff_0, cls_double__inj__on_0, cls_order__less__imp__le_0, cls_abs__le__D1_0, cls_le__divide__eq__number__of1_4, cls_minus__equation__iff_1, clsarity_Complex__Ocomplex__RealVector_Oreal__field, cls_add__leE_1, cls_sum__power2__eq__zero__iff_2, cls_Suc__pred_H_0, cls_Fact_Ofact__0__nat_0, cls_mult__cancel1_0, cls_abs__eq__mult_2, cls_eq__neg__iff__add__eq__0_0, cls_comm__ring__1_Opower__minus1__odd_0, cls_le__imp__inverse__le__neg_0, cls_class__ringb_Osubr0__iff_0, cls_power__diff__rev__if_0, cls_class__semiring_Omul__pwr_0, cls_Suc__mod__eq__add3__mod__number__of_0, clsarity_Int__Oint__Ring__and__Field_Osemiring__1, cls_div__mult__self1__is__id_0, cls_gb__semiring_Opwr__mul_0, cls_monoid__add__class_Oadd__0__right_0, cls_mult__idem_0, cls_add__gr__0_1, cls_ordered__idom_Omult__le__cancel__right1_5, cls_minus1__divide_0, cls_less__divide__eq__number__of_5, cls_zero__less__power__nat__eq__number__of_1, cls_diff__commute_0, cls_ordered__idom_Ozero__le__power2_0, cls_even__nat__equiv__def_1, cls_less__number__of_1, cls_comm__monoid__add_Opower__Suc2_0, cls_division__ring__inverse__diff_0, cls_even__nat__plus__one__div__two_0, cls_le__divide__eq__number__of_4, cls_mod__eq__0__iff_1, cls_le__add1_0, cls_ordered__idom_Omult__less__cancel__left1_4, cls_zero__le__divide__iff_1, cls_power__strict__increasing__iff_0, cls_less__special_I4_J_0, cls_split__div__lemma_2, cls_mod__if_0, cls_div__mult__mult1__if_0, cls_le__less__Suc__eq_1, cls_zero__less__power__nat__eq_2, cls_add__strict__increasing2_0, cls_le__add__diff_0, cls_less__Suc0_1, cls_Suc__n__div__2__gt__zero_0, cls_divide__eq__eq_1, cls_le__divide__eq_5, cls_divide__less__eq__number__of1_1, cls_termination__basic__simps_I2_J_0, cls_semiring__div__class_Omod__div__equality_H_0, cls_le__eq__less__or__eq_0, cls_divide__le__eq__number__of1_0, cls_minus__mult__minus_0, clsarity_nat__Ring__and__Field_Omult__zero, cls_gb__semiring_Osemiring__rules_I29_J_0, clsarity_nat__OrderedGroup_Ocomm__monoid__add, cls_Suc__eq__plus1__left_0, clsarity_Int__Oint__OrderedGroup_Opordered__ab__group__add, cls_zero__less__power__eq__number__of_2, clsarity_Complex__Ocomplex__Ring__and__Field_Ono__zero__divisors, cls_eq__iff__diff__eq__0_0, cls_mod__mult2__eq_0, cls_odd__pos_0, cls_ordered__idom_Opower2__less__0_0, cls_inverse__numeral__1_0, cls_mult__divide__mult__cancel__left_0, cls_mult__le__less__imp__less_0, cls_natgb_Oadd__mul__solve_1, cls_divide__less__0__1__iff_1, clsarity_Int__Oint__Ring__and__Field_Oordered__semiring, cls_ordered__idom_Omult__less__cancel__left2_3, cls_le__divide__eq__number__of1_8, cls_power__Suc__less_0, cls_divide__less__eq__number__of_5, cls_square__eq__iff_0, cls_zero__less__divide__1__iff_0, clsarity_nat__Orderings_Oorder, cls_ordered__idom_Ozero__less__power2_1, cls_order__less__le_1, cls_mult__le__0__iff_3, cls_add__numeral__0__right_0, cls_gb__semiring_Osemiring__rules_I11_J_0, cls_mult__minus1__right_0, cls_xt1_I7_J_0, cls_power2__minus_0, cls_inverse__nonpositive__iff__nonpositive_1, cls_nat__mult__less__cancel1_0, cls_nat__mult__eq__cancel__disj_0, cls_mult__neg__neg_0, cls_zero__less__power__eq__number__of_5, cls_linorder__not__less_1, cls_ordered__idom_Osum__power2__eq__zero__iff_1, cls_divide_Ominus_0, cls_zero__less__Suc_0, cls_mult__less__cancel__left__neg_0, cls_class__semiring_Osemiring__rules_I4_J_0, clsarity_Complex__Ocomplex__Int_Onumber__ring, cls_not__one__le__zero_0, cls_le__neq__implies__less_0, cls_le__add__iff2_0, cls_mult__less__le__imp__less_0, cls_nat__less__le_1, cls_zero__le__power2_0, cls_ordered__idom_Osgn__0__0_0, cls_less__divide__eq__number__of1_5, cls_ordered__idom_Oodd__power__less__zero_0, cls_mult__left_Ozero_0, cls_add__le__cancel__left_0, clsarity_nat__Ring__and__Field_Oordered__semiring__strict, clsarity_Complex__Ocomplex__Ring__and__Field_Ocomm__semiring, cls_rel__simps_I48_J_0, cls_ordered__idom_Omult__le__cancel__left2_3, cls_power__decreasing_0, clsarity_Int__Oint__OrderedGroup_Ocomm__monoid__mult, cls_mod__self_0, cls_gb__semiring_Osemiring__rules_I23_J_0, cls_monoid__mult_Opower3__eq__cube_0, cls_less__eqI_1, cls_zless__linear_0, cls_zero__le__divide__1__iff_0, cls_less__divide__eq_9, cls_divide__less__0__iff_4, cls_split__mult__neg__le_1, cls_nat__eq__add__iff2_1, clsarity_Complex__Ocomplex__Ring__and__Field_Osemiring__0, cls_comm__monoid__add_Opower2__eq__square_0, cls_power__mono_0, cls_le__refl_0, cls_neg__less__iff__less_0, cls_even__less__0__iff_0, cls_le__SucI_0, cls_less__nat__number__of_2, cls_mult__left_Oadd_0, cls_comp__arith_I110_J_0, cls_rel__simps_I46_J_0, cls_divide__le__eq__number__of_3, cls_not__less__eq__eq_0, clsarity_Int__Oint__Ring__and__Field_Oordered__idom, cls_power__0__left_1, cls_power__le__zero__eq__number__of_3, cls_gb__semiring_Osemiring__rules_I15_J_0, cls_le__divide__eq__number__of_9, cls_sum__squares__le__zero__iff_0, cls_less__divide__eq__number__of_9, cls_abs__triangle__ineq4_0, cls_mod__Suc__eq__Suc__mod_0, cls_iszero__0_0, cls_less__2__cases_0, cls_add__le__cancel__right_0, cls_mult__right_Odiff_0, cls_le__divide__eq__number__of_0, cls_gb__semiring_Osemiring__rules_I1_J_0, cls_class__semiring_Ogb__semiring__axioms_H_0, cls_class__semiring_Osemiring__rules_I13_J_0, cls_mult__cancel2_1, cls_mult__strict__left__mono__neg_0, cls_mult__eq__1__iff_2, clsarity_Complex__Ocomplex__Ring__and__Field_Oring__no__zero__divisors, cls_abs__of__neg_0, cls_abs__less__iff_1, cls_mult__less__cancel__left__pos_0, cls_ordered__idom_Omult__less__cancel__right2_2, cls_Suc__div__eq__add3__div__number__of_0, cls_nat__add__commute_0, cls_Deriv_Oadd__diff__add_0, cls_class__semiring_Osemiring__rules_I22_J_0, cls_div__if_1, cls_le__number__of__eq__not__less_0, cls_right__inverse__eq_1, cls_add__diff__inverse_0, cls_divide__le__eq__number__of_6, cls_less__divide__eq__number__of1_1, cls_rel__simps_I43_J_0, cls_abs__diff__less__iff_0, cls_zero__le__mult__iff_2, cls_rel__simps_I6_J_0, cls_abs__power2_0, cls_mod__mult__distrib_0, cls_numeral__1__eq__Suc__0_0, cls_divide__le__eq_1, cls_div__Suc__eq__div__add3_0, cls_divide__le__0__1__iff_0
% 82.58/34.72  % No. of inferences in proof: 3641
% 82.58/34.72  % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 23829 ms resp. 22270 ms w/o parsing
% 90.20/36.16  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 90.29/36.20  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------