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Leo-III---1.8.0.THM-Ref.s

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%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : SWW204+1 : TPTP v9.3.1. Released v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39

% Computer : n019.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 09:09:38 AM UTC 2026

% Result   : Theorem 62.42s 32.50s
% Output   : Refutation 66.80s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      : 1260
% Syntax   : Number of formulae    : 2522 ( 832 unt;   0 typ;   0 def)
%            Number of atoms       : 6324 (1629 equ;   0 cnn)
%            Maximal formula atoms :   26 (   2 avg)
%            Number of connectives : 42229 ( 312   ~; 150   |; 503   &;38115   @)
%                                         ( 221 <=>;2928  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (   8 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  124 ( 122 usr;  12 con; 0-5 aty)
%            Number of variables   : 5454 (   0   ^;5352   !; 102   ?;5454   :)

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

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

thf(c_Orderings_Oord__class_Oless__eq_decl,type,
    c_Orderings_Oord__class_Oless__eq: $i > $i > $i ).

thf(tc_RealDef_Oreal_decl,type,
    tc_RealDef_Oreal: $i ).

thf(c_Groups_Ozero__class_Ozero_decl,type,
    c_Groups_Ozero__class_Ozero: $i > $i ).

thf(v_r_decl,type,
    v_r: $i ).

thf(c_SEQ_Osubseq_decl,type,
    c_SEQ_Osubseq: $i > $o ).

thf(v_ga_____decl,type,
    v_ga____: $i ).

thf(v_f_____decl,type,
    v_f____: $i ).

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

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

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

thf(c_Groups_Oone__class_Oone_decl,type,
    c_Groups_Oone__class_Oone: $i > $i ).

thf(c_Groups_Otimes__class_Otimes_decl,type,
    c_Groups_Otimes__class_Otimes: $i > $i ).

thf(c_Orderings_Oord__class_Oless_decl,type,
    c_Orderings_Oord__class_Oless: $i > $i > $i ).

thf(tc_Nat_Onat_decl,type,
    tc_Nat_Onat: $i ).

thf(c_Nat_OSuc_decl,type,
    c_Nat_OSuc: $i > $i ).

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

thf(c_Complex_Orcis_decl,type,
    c_Complex_Orcis: $i > $i > $i ).

thf(c_RealDef_Oreal_decl,type,
    c_RealDef_Oreal: $i > $i > $i ).

thf(class_Groups_Ocancel__semigroup__add_decl,type,
    class_Groups_Ocancel__semigroup__add: $i > $o ).

thf(class_Rings_Oring_decl,type,
    class_Rings_Oring: $i > $o ).

thf(c_Groups_Oplus__class_Oplus_decl,type,
    c_Groups_Oplus__class_Oplus: $i > $i ).

thf(c_Groups_Ominus__class_Ominus_decl,type,
    c_Groups_Ominus__class_Ominus: $i > $i > $i > $i ).

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

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

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

thf(class_Rings_Oordered__cancel__semiring_decl,type,
    class_Rings_Oordered__cancel__semiring: $i > $o ).

thf(class_Rings_Olinordered__comm__semiring__strict_decl,type,
    class_Rings_Olinordered__comm__semiring__strict: $i > $o ).

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

thf(class_RealVector_Oreal__normed__vector_decl,type,
    class_RealVector_Oreal__normed__vector: $i > $o ).

thf(c_Groups_Oabs__class_Oabs_decl,type,
    c_Groups_Oabs__class_Oabs: $i > $i > $i ).

thf(c_RealVector_Onorm__class_Onorm_decl,type,
    c_RealVector_Onorm__class_Onorm: $i > $i > $i ).

thf(class_Groups_Ocomm__monoid__add_decl,type,
    class_Groups_Ocomm__monoid__add: $i > $o ).

thf(class_Groups_Omonoid__add_decl,type,
    class_Groups_Omonoid__add: $i > $o ).

thf(tc_Polynomial_Opoly_decl,type,
    tc_Polynomial_Opoly: $i > $i ).

thf(class_Orderings_Oord_decl,type,
    class_Orderings_Oord: $i > $o ).

thf(class_Rings_Olinordered__semidom_decl,type,
    class_Rings_Olinordered__semidom: $i > $o ).

thf(class_Groups_Oone_decl,type,
    class_Groups_Oone: $i > $o ).

thf(class_Groups_Oab__group__add_decl,type,
    class_Groups_Oab__group__add: $i > $o ).

thf(class_Rings_Ocomm__semiring__0_decl,type,
    class_Rings_Ocomm__semiring__0: $i > $o ).

thf(class_Rings_Osemiring_decl,type,
    class_Rings_Osemiring: $i > $o ).

thf(class_Groups_Omonoid__mult_decl,type,
    class_Groups_Omonoid__mult: $i > $o ).

thf(c_Complex_OIm_decl,type,
    c_Complex_OIm: $i ).

thf(c_Transcendental_Osin_decl,type,
    c_Transcendental_Osin: $i > $i ).

thf(class_Groups_Ominus_decl,type,
    class_Groups_Ominus: $i > $o ).

thf(c_Complex_ORe_decl,type,
    c_Complex_ORe: $i ).

thf(c_Transcendental_Ocos_decl,type,
    c_Transcendental_Ocos: $i > $i ).

thf(class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_decl,type,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct: $i > $o ).

thf(class_Groups_Oordered__ab__semigroup__add_decl,type,
    class_Groups_Oordered__ab__semigroup__add: $i > $o ).

thf(c_SEQ_OBseq_decl,type,
    c_SEQ_OBseq: $i > $i > $o ).

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

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

thf(class_Rings_Olinordered__idom_decl,type,
    class_Rings_Olinordered__idom: $i > $o ).

thf(class_Rings_Oordered__comm__semiring_decl,type,
    class_Rings_Oordered__comm__semiring: $i > $o ).

thf(class_Rings_Oordered__semiring_decl,type,
    class_Rings_Oordered__semiring: $i > $o ).

thf(class_Groups_Ogroup__add_decl,type,
    class_Groups_Ogroup__add: $i > $o ).

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

thf(class_Rings_Ocomm__semiring__1_decl,type,
    class_Rings_Ocomm__semiring__1: $i > $o ).

thf(class_Rings_Ocomm__semiring_decl,type,
    class_Rings_Ocomm__semiring: $i > $o ).

thf(c_RComplete_Onatceiling_decl,type,
    c_RComplete_Onatceiling: $i > $i ).

thf(class_Groups_Oordered__ab__group__add__abs_decl,type,
    class_Groups_Oordered__ab__group__add__abs: $i > $o ).

thf(class_Rings_Oring__1__no__zero__divisors_decl,type,
    class_Rings_Oring__1__no__zero__divisors: $i > $o ).

thf(class_Groups_Oab__semigroup__add_decl,type,
    class_Groups_Oab__semigroup__add: $i > $o ).

thf(class_Rings_Olinordered__ring_decl,type,
    class_Rings_Olinordered__ring: $i > $o ).

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

thf(tc_fun_decl,type,
    tc_fun: $i > $i > $i ).

thf(class_Groups_Oordered__ab__semigroup__add__imp__le_decl,type,
    class_Groups_Oordered__ab__semigroup__add__imp__le: $i > $o ).

thf(class_Rings_Oidom_decl,type,
    class_Rings_Oidom: $i > $o ).

thf(class_Rings_Ocomm__ring_decl,type,
    class_Rings_Ocomm__ring: $i > $o ).

thf(class_Rings_Olinordered__semiring__1_decl,type,
    class_Rings_Olinordered__semiring__1: $i > $o ).

thf(class_Rings_Omult__zero_decl,type,
    class_Rings_Omult__zero: $i > $o ).

thf(class_Rings_Olinordered__semiring__1__strict_decl,type,
    class_Rings_Olinordered__semiring__1__strict: $i > $o ).

thf(c_SEQ_Omonoseq_decl,type,
    c_SEQ_Omonoseq: $i > $i > $o ).

thf(v_s_decl,type,
    v_s: $i ).

thf(c_Complex_Ocomplex_OComplex_decl,type,
    c_Complex_Ocomplex_OComplex: $i > $i > $i ).

thf(c_Complex_Ocis_decl,type,
    c_Complex_Ocis: $i > $i ).

thf(class_Rings_Ocomm__ring__1_decl,type,
    class_Rings_Ocomm__ring__1: $i > $o ).

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

thf(c_SEQ_Odecseq_decl,type,
    c_SEQ_Odecseq: $i > $i > $o ).

thf(c_Transcendental_Oln_decl,type,
    c_Transcendental_Oln: $i > $i ).

thf(c_RComplete_Onatfloor_decl,type,
    c_RComplete_Onatfloor: $i > $i ).

thf(class_Groups_Ocancel__comm__monoid__add_decl,type,
    class_Groups_Ocancel__comm__monoid__add: $i > $o ).

thf(class_Rings_Olinordered__semiring__strict_decl,type,
    class_Rings_Olinordered__semiring__strict: $i > $o ).

thf(class_Groups_Oordered__comm__monoid__add_decl,type,
    class_Groups_Oordered__comm__monoid__add: $i > $o ).

thf(class_Groups_Oab__semigroup__mult_decl,type,
    class_Groups_Oab__semigroup__mult: $i > $o ).

thf(class_Groups_Olinordered__ab__group__add_decl,type,
    class_Groups_Olinordered__ab__group__add: $i > $o ).

thf(c_NthRoot_Oroot_decl,type,
    c_NthRoot_Oroot: $i > $i > $i ).

thf(class_RealVector_Oreal__normed__div__algebra_decl,type,
    class_RealVector_Oreal__normed__div__algebra: $i > $o ).

thf(class_Rings_Oring__1_decl,type,
    class_Rings_Oring__1: $i > $o ).

thf(c_Fun_Ocomp_decl,type,
    c_Fun_Ocomp: $i > $i > $i > $i > $i > $i ).

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

thf(class_Fields_Ofield_decl,type,
    class_Fields_Ofield: $i > $o ).

thf(class_Groups_Ocomm__monoid__mult_decl,type,
    class_Groups_Ocomm__monoid__mult: $i > $o ).

thf(class_Rings_Ozero__neq__one_decl,type,
    class_Rings_Ozero__neq__one: $i > $o ).

thf(class_Rings_Ono__zero__divisors_decl,type,
    class_Rings_Ono__zero__divisors: $i > $o ).

thf(class_Groups_Oordered__ab__group__add_decl,type,
    class_Groups_Oordered__ab__group__add: $i > $o ).

thf(class_Groups_Ocancel__ab__semigroup__add_decl,type,
    class_Groups_Ocancel__ab__semigroup__add: $i > $o ).

thf(class_Rings_Osemiring__0_decl,type,
    class_Rings_Osemiring__0: $i > $o ).

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

thf(class_Groups_Ozero_decl,type,
    class_Groups_Ozero: $i > $o ).

thf(class_Rings_Olinordered__semiring_decl,type,
    class_Rings_Olinordered__semiring: $i > $o ).

thf(class_Rings_Olinordered__ring__strict_decl,type,
    class_Rings_Olinordered__ring__strict: $i > $o ).

thf(class_Rings_Oordered__ring__abs_decl,type,
    class_Rings_Oordered__ring__abs: $i > $o ).

thf(class_Rings_Oordered__ring_decl,type,
    class_Rings_Oordered__ring: $i > $o ).

thf(class_RealVector_Oreal__normed__algebra__1_decl,type,
    class_RealVector_Oreal__normed__algebra__1: $i > $o ).

thf(class_Rings_Odivision__ring_decl,type,
    class_Rings_Odivision__ring: $i > $o ).

thf(c_SEQ_Oincseq_decl,type,
    c_SEQ_Oincseq: $i > $i > $o ).

thf(class_Fields_Ofield__inverse__zero_decl,type,
    class_Fields_Ofield__inverse__zero: $i > $o ).

thf(class_Rings_Oring__no__zero__divisors_decl,type,
    class_Rings_Oring__no__zero__divisors: $i > $o ).

thf(class_Groups_Oordered__cancel__ab__semigroup__add_decl,type,
    class_Groups_Oordered__cancel__ab__semigroup__add: $i > $o ).

thf(c_Complex_Oexpi_decl,type,
    c_Complex_Oexpi: $i > $i ).

thf(c_Complex_Oarg_decl,type,
    c_Complex_Oarg: $i > $i ).

thf(c_COMBK_decl,type,
    c_COMBK: $i > $i > $i > $i ).

thf(c_Rings_Oinverse__class_Odivide_decl,type,
    c_Rings_Oinverse__class_Odivide: $i > $i > $i > $i ).

thf(c_Complex_Ocomplex_Ocomplex__case_decl,type,
    c_Complex_Ocomplex_Ocomplex__case: $i > $i > $i > $i ).

thf(c_Nat__Transfer_Otsub_decl,type,
    c_Nat__Transfer_Otsub: $i > $i > $i ).

thf(c_Nat_Osize__class_Osize_decl,type,
    c_Nat_Osize__class_Osize: $i > $i > $i ).

thf(class_Lattices_Oab__semigroup__idem__mult_decl,type,
    class_Lattices_Oab__semigroup__idem__mult: $i > $o ).

thf(c_Fundamental__Theorem__Algebra__Mirabelle_Opsize_decl,type,
    c_Fundamental__Theorem__Algebra__Mirabelle_Opsize: $i > $i > $i ).

thf(c_Complex_Ocomplex_Ocomplex__size_decl,type,
    c_Complex_Ocomplex_Ocomplex__size: $i > $i ).

thf(c_Complex_Ocomplex_Ocomplex__rec_decl,type,
    c_Complex_Ocomplex_Ocomplex__rec: $i > $i > $i > $i ).

thf(605,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__less__imp__less__base) ).

thf(3021,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(59,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__assoc) ).

thf(1411,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(143,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
     => ( ( B @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__zmult__eq__1) ).

thf(1633,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
     => ( ( B @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(834,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ? [B: $i] :
          ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reals__Archimedean6) ).

thf(3798,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ? [B: $i] :
          ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(887,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__0) ).

thf(4022,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(728,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__abs__iff) ).

thf(3434,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( ( A
           != ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A
           != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(384,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__Suc__less__one) ).

thf(2331,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(1093,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__add__mult__distrib) ).

thf(4728,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).

thf(111,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semidom) ).

thf(1549,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(365,axiom,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__add) ).

thf(2280,plain,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(91,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) ).

thf(1501,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(938,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_SEQ_OBseq ) )
      <=> ? [C: $i] :
          ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Bseq__iff) ).

thf(4201,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_SEQ_OBseq ) ) )
        & ( ( A @ ( B @ c_SEQ_OBseq ) )
         => ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(1038,axiom,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__ge__zero) ).

thf(4560,plain,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).

thf(172,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Omult__zero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Omult__zero) ).

thf(1718,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Omult__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(477,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1) ).

thf(2612,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(907,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_RComplete_Onatceiling @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__mono) ).

thf(4094,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_RComplete_Onatceiling @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(83,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__right__cancel) ).

thf(1469,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( C = A )
         => ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(43,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Transcendental_Ocos )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__add) ).

thf(1369,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Transcendental_Ocos )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(1130,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE) ).

thf(4835,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).

thf(199,axiom,
    tc_Int_Oint @ class_Groups_Oab__group__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__group__add) ).

thf(1823,plain,
    tc_Int_Oint @ class_Groups_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(374,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Power_Opower )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__0) ).

thf(2306,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Power_Opower )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(1012,axiom,
    ! [A: $i] : ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__add__one__gt__zero) ).

thf(4464,plain,
    ! [A: $i] : ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).

thf(616,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__eq__zero__iff) ).

thf(3053,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(910,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I2_J) ).

thf(4101,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(388,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__semiring) ).

thf(2336,plain,
    tc_Nat_Onat @ class_Rings_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(322,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__cancel__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring) ).

thf(2131,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(74,axiom,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(1454,plain,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(299,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Ocis ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DeMoivre) ).

thf(2072,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Ocis ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(1245,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__le__less__linear) ).

thf(5329,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1245]) ).

thf(791,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add1__zle__eq) ).

thf(3661,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(716,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq3) ).

thf(3399,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(270,axiom,
    tc_Nat_Onat @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Divides_Osemiring__div) ).

thf(1995,plain,
    tc_Nat_Onat @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(865,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__mult1) ).

thf(3930,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(927,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel2) ).

thf(4169,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(569,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__cancel__iff) ).

thf(2921,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(965,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__leI) ).

thf(4290,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(736,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__cancel__left) ).

thf(3477,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(799,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__add__left__mono) ).

thf(3698,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(434,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict) ).

thf(2463,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(715,axiom,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__zero__def) ).

thf(3397,plain,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(25,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__group__add) ).

thf(1315,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(473,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(2595,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(420,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1) ).

thf(2427,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(875,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__strict__right__mono) ).

thf(3969,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(1208,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__iff2) ).

thf(5205,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1208]) ).

thf(886,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psize__eq__0__iff) ).

thf(4015,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(1142,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__funE) ).

thf(4877,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).

thf(197,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__group__add) ).

thf(1818,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(15,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Oadd) ).

thf(1295,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(163,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_RComplete_Onatceiling )
    = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__one) ).

thf(1686,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_RComplete_Onatceiling )
    = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(926,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__mult2) ).

thf(4166,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(240,axiom,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lessI) ).

thf(1924,plain,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(902,axiom,
    tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).

thf(4075,plain,
    tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(820,axiom,
    tc_RealDef_Oreal @ class_Groups_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__ab__semigroup__add) ).

thf(3762,plain,
    tc_RealDef_Oreal @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(194,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel__less) ).

thf(1808,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(748,axiom,
    tc_RealDef_Oreal @ class_Fields_Ofield__inverse__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Fields_Ofield__inverse__zero) ).

thf(3509,plain,
    tc_RealDef_Oreal @ class_Fields_Ofield__inverse__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(355,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Oone__class_Oone )
       != ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__neq__zero) ).

thf(2248,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Oone__class_Oone )
       != ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(109,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) ).

thf(1544,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(632,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ? [B: $i] :
          ( ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reals__Archimedean6a) ).

thf(3108,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ? [B: $i] :
          ( ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(315,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ ( c_Complex_ORe @ hAPP ) )
        = ( A @ ( c_Complex_ORe @ hAPP ) ) )
     => ( ( ( B @ ( c_Complex_OIm @ hAPP ) )
          = ( A @ ( c_Complex_OIm @ hAPP ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__equality) ).

thf(2114,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ ( c_Complex_ORe @ hAPP ) )
        = ( A @ ( c_Complex_ORe @ hAPP ) ) )
     => ( ( ( B @ ( c_Complex_OIm @ hAPP ) )
          = ( A @ ( c_Complex_OIm @ hAPP ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(166,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc )
        = ( A @ c_Nat_OSuc ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Oinject) ).

thf(1692,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
       => ( ( B @ c_Nat_OSuc )
          = ( A @ c_Nat_OSuc ) ) )
      & ( ( ( B @ c_Nat_OSuc )
          = ( A @ c_Nat_OSuc ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(1254,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__mult__ineq) ).

thf(5348,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1254]) ).

thf(1021,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
        = ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) )
    <=> ( ( A
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
        & ( B
          = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Complex__eq__1) ).

thf(4487,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) )
       => ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
          = ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) ) )
      & ( ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
          = ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) )
       => ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).

thf(1018,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1__strict )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_convex__bound__lt) ).

thf(4479,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1__strict )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).

thf(302,axiom,
    tc_Nat_Onat @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ono__zero__divisors) ).

thf(2077,plain,
    tc_Nat_Onat @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(1158,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__le__zero__iff) ).

thf(4931,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).

thf(375,axiom,
    tc_Int_Oint @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Olinorder) ).

thf(2309,plain,
    tc_Int_Oint @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(989,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__nonpos__nonpos) ).

thf(4376,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(316,axiom,
    tc_Nat_Onat @ class_Groups_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) ).

thf(2117,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(57,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring__1__no__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors) ).

thf(1406,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(311,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
     != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_odd__nonzero) ).

thf(2104,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
     != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(693,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__div__cancel1) ).

thf(3317,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(1196,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Ocomplex_Ocomplex__size )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex_Osize_I1_J) ).

thf(5168,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Ocomplex_Ocomplex__size )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1196]) ).

thf(1024,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatceiling )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__add__one) ).

thf(4500,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatceiling )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).

thf(837,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__imp__zdiv__nonneg__iff) ).

thf(3813,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(1234,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__increasing) ).

thf(5292,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1234]) ).

thf(408,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__assoc) ).

thf(2395,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(566,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__cancel__left) ).

thf(2908,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(295,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).

thf(2062,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(762,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Nat_OSuc @ c_NthRoot_Oroot ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__pos) ).

thf(3549,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Nat_OSuc @ c_NthRoot_Oroot ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(712,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__le__mono) ).

thf(3377,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(612,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__of__nonneg) ).

thf(3040,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(469,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring__no__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors) ).

thf(2590,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(1236,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__le__zero__iff) ).

thf(5297,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1236]) ).

thf(309,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__mult__def) ).

thf(2099,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(1213,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Opos__bounded) ).

thf(5224,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1213]) ).

thf(128,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add) ).

thf(1591,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(1225,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( B @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hBOOL )
            & ! [D: $i] :
                ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ~ ( D @ ( B @ hAPP ) @ hBOOL ) )
            & ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ex__least__nat__less) ).

thf(5265,plain,
    ! [A: $i,B: $i] :
      ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( B @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hBOOL )
            & ! [D: $i] :
                ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ~ ( D @ ( B @ hAPP ) @ hBOOL ) )
            & ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1225]) ).

thf(942,axiom,
    ( ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__one__def) ).

thf(4215,plain,
    ( ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(259,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).

thf(1971,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(1055,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Ononneg__bounded) ).

thf(4614,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).

thf(86,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__diff) ).

thf(1486,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(911,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff) ).

thf(4105,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(341,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__left__cancel) ).

thf(2195,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B = A )
         => ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(502,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__Suc1) ).

thf(2686,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(1045,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring) ).

thf(4575,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).

thf(639,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_Ocomplex__case ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex_Osimps_I2_J) ).

thf(3135,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_Ocomplex__case ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(523,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc__eq) ).

thf(2740,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(1232,axiom,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__le__one) ).

thf(5288,plain,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1232]) ).

thf(428,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__poly__add__left) ).

thf(2448,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(962,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__less__imp__less) ).

thf(4279,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

thf(827,axiom,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__zero) ).

thf(3781,plain,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(1203,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__le__imp__le) ).

thf(5192,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1203]) ).

thf(1105,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__of__pos) ).

thf(4762,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).

thf(680,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__mono__neg) ).

thf(3276,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(1183,axiom,
    ! [A: $i] :
      ( ( A @ c_SEQ_Osubseq )
    <=> ! [B: $i,C: $i] :
          ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subseq__def) ).

thf(5120,plain,
    ! [A: $i] :
      ( ( ! [B: $i,C: $i] :
            ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ c_SEQ_Osubseq ) )
      & ( ( A @ c_SEQ_Osubseq )
       => ! [B: $i,C: $i] :
            ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).

thf(660,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__right__mono__neg) ).

thf(3217,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(833,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__lt__1__iff) ).

thf(3794,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(134,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__less__add1) ).

thf(1608,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(1063,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__eq__eq) ).

thf(4633,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(614,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq2) ).

thf(3049,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(814,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq4) ).

thf(3745,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(1097,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__antisym) ).

thf(4739,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).

thf(913,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__zero__iff) ).

thf(4110,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(219,axiom,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__mult) ).

thf(1867,plain,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(336,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1) ).

thf(2184,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(378,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__1) ).

thf(2316,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(731,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) )
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__eq__zero) ).

thf(3446,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) )
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) )
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(1189,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I3_J) ).

thf(5146,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).

thf(222,axiom,
    tc_Int_Oint @ class_Orderings_Oord,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Oord) ).

thf(1872,plain,
    tc_Int_Oint @ class_Orderings_Oord,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(513,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ c_RComplete_Onatfloor )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__eq) ).

thf(2712,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ c_RComplete_Onatfloor )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(1068,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__nonneg__nonneg) ).

thf(4650,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(483,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel) ).

thf(2627,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B = A )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(673,axiom,
    tc_RealDef_Oreal @ class_Groups_Olinordered__ab__group__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Olinordered__ab__group__add) ).

thf(3253,plain,
    tc_RealDef_Oreal @ class_Groups_Olinordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(267,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring__strict) ).

thf(1988,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(1195,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv1) ).

thf(5161,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( B = A )
           => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).

thf(737,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__le__mono) ).

thf(3481,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(252,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__mult__distrib2) ).

thf(1954,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(553,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
    <=> ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        | ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        | ( A
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__eq__0__iff) ).

thf(2844,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( A
            = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( A
            = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(40,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__eq__if) ).

thf(1354,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(327,axiom,
    tc_Nat_Onat @ class_Groups_Ozero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ozero) ).

thf(2146,plain,
    tc_Nat_Onat @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(1160,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ! [C: $i,D: $i] :
            ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_monoI2) ).

thf(4941,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ! [C: $i,D: $i] :
            ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).

thf(343,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__assoc) ).

thf(2216,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(703,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__le__0__iff) ).

thf(3350,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(53,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__less__cancel1) ).

thf(1391,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(1056,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__imp__zdiv__pos__iff) ).

thf(4618,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).

thf(243,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__by__0) ).

thf(1935,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(239,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__one) ).

thf(1921,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(561,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ c_RComplete_Onatfloor @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__natfloor__eq) ).

thf(2886,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ c_RComplete_Onatfloor @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ c_RComplete_Onatfloor @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(175,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__less__Suc__eq) ).

thf(1724,plain,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B = A )
         => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(42,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Ocos )
      = ( B @ c_Transcendental_Osin @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Transcendental_Ocos @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__diff2) ).

thf(1366,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Ocos )
      = ( B @ c_Transcendental_Osin @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Transcendental_Ocos @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(5,axiom,
    ! [A: $i] :
      ( ( A @ class_Power_Opower )
     => ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ c_Groups_Otimes__class_Otimes @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Power_Opower_Opower ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__power__power) ).

thf(1265,plain,
    ! [A: $i] :
      ( ( A @ class_Power_Opower )
     => ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ c_Groups_Otimes__class_Otimes @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Power_Opower_Opower ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(449,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring) ).

thf(2531,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(993,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_SEQ_OBseq ) )
      <=> ? [C: $i] :
            ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Bseq__def) ).

thf(4386,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ? [C: $i] :
              ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ c_SEQ_OBseq ) ) )
        & ( ( A @ ( B @ c_SEQ_OBseq ) )
         => ? [C: $i] :
              ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(785,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = A )
      <=> ( B
          = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__diff__is__add) ).

thf(3634,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
            = A ) )
        & ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
            = A )
         => ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(1219,axiom,
    tc_RealDef_Oreal @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__semigroup__mult) ).

thf(5252,plain,
    tc_RealDef_Oreal @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1219]) ).

thf(822,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Nat_Osize__class_Osize ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex_Osize_I2_J) ).

thf(3766,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Nat_Osize__class_Osize ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(881,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__real__le) ).

thf(3994,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(860,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__less__mono) ).

thf(3915,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(454,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring) ).

thf(2548,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(190,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Ozero__neq__one )
        & ( C @ class_Rings_Ono__zero__divisors )
        & ( C @ class_Rings_Omult__zero )
        & ( C @ class_Power_Opower ) )
     => ( ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__eq__0__iff) ).

thf(1782,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Ozero__neq__one )
        & ( C @ class_Rings_Ono__zero__divisors )
        & ( C @ class_Rings_Omult__zero )
        & ( C @ class_Power_Opower ) )
     => ( ( ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(549,axiom,
    ! [A: $i] :
      ( ( ( A @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
     => ( ( A @ c_Transcendental_Osin )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__one__sin__zero) ).

thf(2825,plain,
    ! [A: $i] :
      ( ( ( A @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
     => ( ( A @ c_Transcendental_Osin )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(103,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Transcendental_Oln )
        = ( B @ c_Transcendental_Oln @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__realpow) ).

thf(1529,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Transcendental_Oln )
        = ( B @ c_Transcendental_Oln @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(1074,axiom,
    ( c_Complex_Oii
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_i__def) ).

thf(4665,plain,
    ( c_Complex_Oii
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(976,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__less__def) ).

thf(4322,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(1256,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl3) ).

thf(5352,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1256]) ).

thf(1250,axiom,
    tc_RealDef_Oreal @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__semigroup__add) ).

thf(5342,plain,
    tc_RealDef_Oreal @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1250]) ).

thf(228,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__mult) ).

thf(1887,plain,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(653,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos) ).

thf(3195,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(519,axiom,
    ! [A: $i,B: $i] : ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_metric__bound__lemma) ).

thf(2735,plain,
    ! [A: $i,B: $i] : ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(493,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__group__add__abs ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__group__add__abs) ).

thf(2667,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__group__add__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(1083,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) )
      = ( B @ ( A @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__divide__square__eq) ).

thf(4692,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) )
      = ( B @ ( A @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).

thf(744,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__right__mono) ).

thf(3495,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(769,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__mono) ).

thf(3574,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(28,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__add) ).

thf(1323,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(1109,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) ).

thf(4778,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).

thf(204,axiom,
    tc_Complex_Ocomplex @ class_Power_Opower,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Power_Opower) ).

thf(1832,plain,
    tc_Complex_Ocomplex @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(320,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Im__mult) ).

thf(2126,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(395,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Osin )
      = ( B @ c_Transcendental_Ocos @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Transcendental_Osin @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__diff2) ).

thf(2356,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Osin )
      = ( B @ c_Transcendental_Ocos @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Transcendental_Osin @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(953,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ ( C @ c_NthRoot_Oroot ) )
          = ( A @ ( C @ c_NthRoot_Oroot ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__eq__iff) ).

thf(4243,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B = A )
         => ( ( B @ ( C @ c_NthRoot_Oroot ) )
            = ( A @ ( C @ c_NthRoot_Oroot ) ) ) )
        & ( ( ( B @ ( C @ c_NthRoot_Oroot ) )
            = ( A @ ( C @ c_NthRoot_Oroot ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(671,axiom,
    ( ( c_Complex_Oii @ ( c_Complex_OIm @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Im__i) ).

thf(3248,plain,
    ( ( c_Complex_Oii @ ( c_Complex_OIm @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(951,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE__linordered__idom) ).

thf(4238,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(877,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__nonpos__pos__le0) ).

thf(3976,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(383,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ c_RComplete_Onatfloor )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__real__of__nat) ).

thf(2328,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ c_RComplete_Onatfloor )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(1237,axiom,
    tc_RealDef_Oreal @ class_Rings_Ocomm__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__ring) ).

thf(5306,plain,
    tc_RealDef_Oreal @ class_Rings_Ocomm__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1237]) ).

thf(1150,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ( ? [D: $i] :
            ( ( D @ ( C @ hAPP ) @ hBOOL )
            & ( D @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( B @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        | ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
          & ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__div_H) ).

thf(4896,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ? [D: $i] :
              ( ( D @ ( C @ hAPP ) @ hBOOL )
              & ( D @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( B @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
            & ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
       => ( ? [D: $i] :
              ( ( D @ ( C @ hAPP ) @ hBOOL )
              & ( D @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( B @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
            & ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).

thf(191,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( B != A )
          & ( D != C ) )
      <=> ( ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
         != ( B @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_crossproduct__noteq) ).

thf(1792,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
           != ( B @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( ( B != A )
            & ( D != C ) ) )
        & ( ( ( B != A )
            & ( D != C ) )
         => ( ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
           != ( B @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(508,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__SucE) ).

thf(2698,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(181,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__Suc2) ).

thf(1760,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(155,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__Suc__less) ).

thf(1660,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(16,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__comm__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) ).

thf(1298,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(215,axiom,
    tc_Int_Oint @ class_Rings_Osemiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring) ).

thf(1860,plain,
    tc_Int_Oint @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(4,axiom,
    v_f____ @ c_SEQ_Osubseq,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_f_I1_J) ).

thf(1264,plain,
    v_f____ @ c_SEQ_Osubseq,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(978,axiom,
    tc_RealDef_Oreal @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Omult__zero) ).

thf(4334,plain,
    tc_RealDef_Oreal @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(636,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__double__add__iff__zero__less__single__add) ).

thf(3125,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(1190,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff__inverse) ).

thf(5150,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).

thf(1185,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__gr__0) ).

thf(5130,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).

thf(121,axiom,
    tc_Int_Oint @ class_Groups_Olinordered__ab__group__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Olinordered__ab__group__add) ).

thf(1577,plain,
    tc_Int_Oint @ class_Groups_Olinordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(75,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1__no__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors) ).

thf(1455,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(682,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__add__iff1) ).

thf(3280,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(269,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__eq__diff__pred) ).

thf(1992,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(1042,axiom,
    tc_RealDef_Oreal @ class_Rings_Oordered__ring__abs,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__ring__abs) ).

thf(4568,plain,
    tc_RealDef_Oreal @ class_Rings_Oordered__ring__abs,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(78,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1) ).

thf(1462,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(613,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel__le) ).

thf(3043,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(505,axiom,
    ! [A: $i,B: $i] :
      ~ ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__add__less2) ).

thf(2691,plain,
    ! [A: $i,B: $i] :
      ~ ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(1140,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nonneg1__imp__zdiv__pos__iff) ).

thf(4862,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1140]) ).

thf(1206,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__div__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__mult) ).

thf(5199,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__div__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1206]) ).

thf(168,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__diff__conv) ).

thf(1704,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(119,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__mult) ).

thf(1568,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(256,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__self__eq__0) ).

thf(1963,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(955,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__Suc__le) ).

thf(4253,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(916,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__ge__zero) ).

thf(4119,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(752,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__less) ).

thf(3517,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(711,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ c_Nat__Transfer_Otsub ) )
        = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tsub__eq) ).

thf(3374,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ c_Nat__Transfer_Otsub ) )
        = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(1221,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Ozero) ).

thf(5256,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1221]) ).

thf(985,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__zero__right) ).

thf(4358,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(456,axiom,
    tc_Nat_Onat @ class_Rings_Osemiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring) ).

thf(2552,plain,
    tc_Nat_Onat @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(1073,axiom,
    tc_RealDef_Oreal @ class_Rings_Oring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__1) ).

thf(4664,plain,
    tc_RealDef_Oreal @ class_Rings_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(531,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__ge__zero) ).

thf(2767,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(401,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
    <=> ( ( ( B @ ( c_Complex_OIm @ hAPP ) )
          = ( A @ ( c_Complex_OIm @ hAPP ) ) )
        & ( ( B @ ( c_Complex_ORe @ hAPP ) )
          = ( A @ ( c_Complex_ORe @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_expand__complex__eq) ).

thf(2373,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( ( B @ ( c_Complex_OIm @ hAPP ) )
            = ( A @ ( c_Complex_OIm @ hAPP ) ) )
          & ( ( B @ ( c_Complex_ORe @ hAPP ) )
            = ( A @ ( c_Complex_ORe @ hAPP ) ) ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( ( B @ ( c_Complex_OIm @ hAPP ) )
            = ( A @ ( c_Complex_OIm @ hAPP ) ) )
          & ( ( B @ ( c_Complex_ORe @ hAPP ) )
            = ( A @ ( c_Complex_ORe @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(1057,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__asym_H) ).

thf(4622,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(862,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ~ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__not__less__zero) ).

thf(3919,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ~ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(974,axiom,
    tc_RealDef_Oreal @ class_Rings_Odivision__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Odivision__ring) ).

thf(4317,plain,
    tc_RealDef_Oreal @ class_Rings_Odivision__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(627,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = C ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power_Opower_Opower__0) ).

thf(3084,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(572,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__neg__nonpos) ).

thf(2929,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(1078,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1 )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_convex__bound__le) ).

thf(4675,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1 )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).

thf(484,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__add1__eq) ).

thf(2635,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(1169,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__cancel2) ).

thf(5052,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( C = A ) )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).

thf(812,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( B
          = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( A
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__0__iff) ).

thf(3734,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B
            = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( B
            = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(988,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__is__0__eq) ).

thf(4368,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(444,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__le__imp__le__exp) ).

thf(2520,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(517,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( c_Complex_ORe @ hAPP ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re) ).

thf(2730,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( c_Complex_ORe @ hAPP ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(872,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor )
        = ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__add) ).

thf(3951,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor )
        = ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(331,axiom,
    tc_Int_Oint @ class_Rings_Oring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1) ).

thf(2158,plain,
    tc_Int_Oint @ class_Rings_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(146,axiom,
    tc_Int_Oint @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__mult) ).

thf(1641,plain,
    tc_Int_Oint @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(840,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( ( C
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) )
        & ( ( C
            = ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__mult1__if) ).

thf(3835,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( ( C
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) )
        & ( ( C
            = ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(992,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__assoc2) ).

thf(4383,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(371,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__diff__diff) ).

thf(2301,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(196,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__cancel) ).

thf(1815,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(224,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__1) ).

thf(1875,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(1115,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B != A )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__neq__implies__less) ).

thf(4793,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B != A )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).

thf(1027,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          | ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          | ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__0__le__divide__iff) ).

thf(4507,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).

thf(404,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__less__power) ).

thf(2388,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(450,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__0__eq__0) ).

thf(2533,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(431,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult) ).

thf(2456,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(345,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ominus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ominus ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Groups_Ominus) ).

thf(2222,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ominus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ominus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(695,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono2) ).

thf(3322,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(279,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__semidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semidom) ).

thf(2027,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(623,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__pos__pos__trivial) ).

thf(3075,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(306,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__mult) ).

thf(2090,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(801,axiom,
    tc_RealDef_Oreal @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Olinorder) ).

thf(3702,plain,
    tc_RealDef_Oreal @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(755,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ( E @ ( F @ ( G @ ( H @ ( I @ c_Fun_Ocomp ) ) ) ) )
        = ( B @ ( C @ ( G @ ( H @ ( D @ c_Fun_Ocomp ) ) ) ) ) )
     => ( ( A @ ( E @ hAPP ) @ ( F @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__eq__dest) ).

thf(3525,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ( E @ ( F @ ( G @ ( H @ ( I @ c_Fun_Ocomp ) ) ) ) )
        = ( B @ ( C @ ( G @ ( H @ ( D @ c_Fun_Ocomp ) ) ) ) ) )
     => ( ( A @ ( E @ hAPP ) @ ( F @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(1123,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__mult__pos) ).

thf(4815,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).

thf(1193,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_NthRoot_Oroot ) )
        = ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__mult) ).

thf(5156,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_NthRoot_Oroot ) )
        = ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).

thf(536,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__zero__left) ).

thf(2777,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(1222,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__le__mono) ).

thf(5259,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1222]) ).

thf(824,axiom,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_Opos__bounded) ).

thf(3771,plain,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(87,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) ).

thf(1489,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(351,axiom,
    tc_Complex_Ocomplex @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Omult__zero) ).

thf(2237,plain,
    tc_Complex_Ocomplex @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(479,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) )
      = ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__norm__def) ).

thf(2616,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) )
      = ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(248,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring) ).

thf(1946,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(1137,axiom,
    tc_RealDef_Oreal @ class_Rings_Oring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring) ).

thf(4857,plain,
    tc_RealDef_Oreal @ class_Rings_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).

thf(599,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__less__mono) ).

thf(2996,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(1010,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_leI) ).

thf(4454,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).

thf(688,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__mono) ).

thf(3295,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(538,axiom,
    tc_RealDef_Oreal @ class_Rings_Oordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__semiring) ).

thf(2782,plain,
    tc_RealDef_Oreal @ class_Rings_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(142,axiom,
    tc_Int_Oint @ class_Rings_Oordered__comm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__comm__semiring) ).

thf(1632,plain,
    tc_Int_Oint @ class_Rings_Oordered__comm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(968,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__mono1) ).

thf(4297,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(437,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ! [D: $i,E: $i] :
              ( ( A @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( ( B
                  = ( E @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
               => ( D @ ( C @ hAPP ) @ hBOOL ) ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__div) ).

thf(2469,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ! [D: $i,E: $i] :
                ( ( A @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
               => ( ( B
                    = ( E @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                 => ( D @ ( C @ hAPP ) @ hBOOL ) ) ) )
          & ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
       => ( ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ! [D: $i,E: $i] :
                ( ( A @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
               => ( ( B
                    = ( E @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                 => ( D @ ( C @ hAPP ) @ hBOOL ) ) ) )
          & ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(1247,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__imp__le) ).

thf(5333,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1247]) ).

thf(1125,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__commute) ).

thf(4820,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).

thf(732,axiom,
    ! [A: $i] : ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__natfloor) ).

thf(3453,plain,
    ! [A: $i] : ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(429,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__left__imp__eq) ).

thf(2451,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(202,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__1__right) ).

thf(1828,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(570,axiom,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra )
     => ? [B: $i] :
        ! [C: $i,D: $i] : ( B @ ( D @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Obounded) ).

thf(2925,plain,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra )
     => ? [B: $i] :
        ! [C: $i,D: $i] : ( B @ ( D @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(370,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__1__iff) ).

thf(2291,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(920,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__increasing) ).

thf(4134,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(359,axiom,
    tc_Nat_Onat @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ozero__neq__one) ).

thf(2254,plain,
    tc_Nat_Onat @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(159,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_realpow__two__diff) ).

thf(1677,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(453,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
        = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__inject) ).

thf(2540,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
       => ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
          = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) ) )
      & ( ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
          = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(1239,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__eq__diff) ).

thf(5311,plain,
    ! [A: $i,B: $i] :
      ( ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1239]) ).

thf(686,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__neg__neg__trivial) ).

thf(3290,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(864,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ c_NthRoot_Oroot ) )
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__eq__0__iff) ).

thf(3923,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_NthRoot_Oroot ) )
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_NthRoot_Oroot ) )
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(255,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__mult__distrib2) ).

thf(1960,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(123,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__commutes) ).

thf(1581,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(387,axiom,
    tc_Nat_Onat @ class_Groups_Oone,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oone) ).

thf(2335,plain,
    tc_Nat_Onat @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(293,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__strict) ).

thf(2058,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(1072,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__ge__1__iff) ).

thf(4660,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(233,axiom,
    tc_Complex_Ocomplex @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring__0) ).

thf(1896,plain,
    tc_Complex_Ocomplex @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(71,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Olinorder) ).

thf(1447,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(746,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__one) ).

thf(3505,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(534,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) ).

thf(2772,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(697,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__mono) ).

thf(3327,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(524,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self2) ).

thf(2750,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(247,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__0__right) ).

thf(1943,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(1088,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__le__eq) ).

thf(4707,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).

thf(1148,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__self) ).

thf(4892,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).

thf(1060,axiom,
    v_r @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( v_s @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact__096cmod_A_Is_A0_J_A_060_061_Ar_096) ).

thf(4628,plain,
    v_r @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( v_s @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(742,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I4_J) ).

thf(3492,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(652,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__cancel1) ).

thf(3184,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B = A ) )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(714,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc0) ).

thf(3389,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(587,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) ).

thf(2964,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(399,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__commute) ).

thf(2367,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(424,axiom,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__1__right) ).

thf(2437,plain,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__add__iff1) ).

thf(1928,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = A )
         => ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(897,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_Transcendental_Oln )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__one) ).

thf(4055,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_Transcendental_Oln )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(948,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono) ).

thf(4230,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(892,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__diff) ).

thf(4033,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(366,axiom,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Osin @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__cos__squared__add3) ).

thf(2281,plain,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Osin @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(1132,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__less__iff) ).

thf(4842,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).

thf(1211,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring__1__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__1__strict) ).

thf(5220,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring__1__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1211]) ).

thf(446,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ c_RComplete_Onatceiling )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__eq) ).

thf(2523,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ c_RComplete_Onatceiling )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(610,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I10_J) ).

thf(3036,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(138,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).

thf(1622,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(419,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__1__right) ).

thf(2424,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(1251,axiom,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__div__algebra,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra) ).

thf(5343,plain,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__div__algebra,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1251]) ).

thf(324,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Deriv_Oadd__diff__add) ).

thf(2137,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(1242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B != A )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__neq__trans) ).

thf(5321,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B != A )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1242]) ).

thf(972,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( D @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          | ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__left__disj) ).

thf(4307,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( ( ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( D @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
            | ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( D @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
            | ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(706,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__less__trans) ).

thf(3361,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(274,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__lessD1) ).

thf(2002,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(1031,axiom,
    tc_RealDef_Oreal @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Omonoid__mult) ).

thf(4536,plain,
    tc_RealDef_Oreal @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).

thf(668,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ~ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__not__less__zero) ).

thf(3243,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ~ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(780,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__le__one__le) ).

thf(3606,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zpower__zpower) ).

thf(1521,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(749,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( E @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( E @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( B @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__quotient__lemma) ).

thf(3510,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( E @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( E @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( B @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(117,axiom,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__add) ).

thf(1564,plain,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(217,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oidom) ).

thf(1863,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(1198,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( C = A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__eq__imp__eq__base) ).

thf(5174,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( C = A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).

thf(846,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__neg__neg) ).

thf(3857,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(937,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__pos__pos) ).

thf(4199,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(776,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( ( B = A )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__less) ).

thf(3591,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( ( ( B = A )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( B = A )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(1101,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono) ).

thf(4753,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).

thf(165,axiom,
    tc_Nat_Onat @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Omult__zero) ).

thf(1691,plain,
    tc_Nat_Onat @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(880,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__is__0) ).

thf(3984,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(317,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Olinordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add) ).

thf(2118,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Olinordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(1173,axiom,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__Suc__gt__zero) ).

thf(5070,plain,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).

thf(1036,axiom,
    tc_RealDef_Oreal @ class_Orderings_Oord,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Oord) ).

thf(4557,plain,
    tc_RealDef_Oreal @ class_Orderings_Oord,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).

thf(1151,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__linear) ).

thf(4910,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).

thf(220,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_decr__lemma) ).

thf(1868,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(581,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__ge__self) ).

thf(2954,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(151,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I1_J) ).

thf(1651,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(1135,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__less__mult) ).

thf(4853,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).

thf(664,axiom,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A
        = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zabs__less__one__iff) ).

thf(3229,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(1048,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__gt__zero) ).

thf(4589,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).

thf(1061,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__neg__pos__less0) ).

thf(4629,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(464,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) ).

thf(2574,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(1146,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__pos__pos) ).

thf(4888,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).

thf(67,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__ring) ).

thf(1438,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(844,axiom,
    ! [A: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__le__cmod) ).

thf(3852,plain,
    ! [A: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(20,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__add) ).

thf(1306,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(398,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__left) ).

thf(2364,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(334,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).

thf(2164,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(238,axiom,
    ! [A: $i] :
      ( ( A @ c_SEQ_Osubseq )
    <=> ! [B: $i] : ( B @ c_Nat_OSuc @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subseq__Suc__iff) ).

thf(1915,plain,
    ! [A: $i] :
      ( ( ! [B: $i] : ( B @ c_Nat_OSuc @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ c_SEQ_Osubseq ) )
      & ( ( A @ c_SEQ_Osubseq )
       => ! [B: $i] : ( B @ c_Nat_OSuc @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(307,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Osin )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__diff) ).

thf(2093,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Osin )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(903,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__le) ).

thf(4076,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(520,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring) ).

thf(2737,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(22,axiom,
    tc_Nat_Onat @ class_Orderings_Oord,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oord) ).

thf(1311,plain,
    tc_Nat_Onat @ class_Orderings_Oord,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(349,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__strict__right__mono) ).

thf(2232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(1046,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__zero__less__power__iff) ).

thf(4576,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).

thf(891,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__nonneg__neg__le0) ).

thf(4031,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(909,axiom,
    tc_RealDef_Oreal @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Osemiring__0) ).

thf(4100,plain,
    tc_RealDef_Oreal @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(540,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv2) ).

thf(2790,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( B = A )
           => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(273,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__strict__decreasing) ).

thf(2000,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(144,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__zmult__distrib) ).

thf(1636,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(723,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reals__Archimedean4) ).

thf(3418,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(1156,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__zless__mono) ).

thf(4925,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).

thf(413,axiom,
    ? [A: $i] :
      ( ( A @ ( v_f____ @ ( v_s @ ( c_Complex_OIm @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_Fun_Ocomp ) ) ) ) @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Nat_Onat @ c_Fun_Ocomp ) ) ) ) @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) )
      & ( A @ c_SEQ_Osubseq ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact__096EX_Afa_O_Asubseq_Afa_A_G_Amonoseq_A_I_Fn_O_A_IIm_Ao_As_Ao_Af_J_A_Ifa_An_J_J_096) ).

thf(2409,plain,
    ? [A: $i] :
      ( ( A @ ( v_f____ @ ( v_s @ ( c_Complex_OIm @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_Fun_Ocomp ) ) ) ) @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Nat_Onat @ c_Fun_Ocomp ) ) ) ) @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) )
      & ( A @ c_SEQ_Osubseq ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(1013,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__power__less__imp__less) ).

thf(4466,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).

thf(421,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc )
        = ( A @ c_Nat_OSuc ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__inject) ).

thf(2428,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc )
        = ( A @ c_Nat_OSuc ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(99,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I2_J) ).

thf(1519,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(1154,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__order) ).

thf(4920,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).

thf(328,axiom,
    tc_Int_Oint @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Oorder) ).

thf(2147,plain,
    tc_Int_Oint @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(853,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ( B = A )
          | ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__iff__gr__or__eq) ).

thf(3884,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( ( B = A )
            | ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
         => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B = A )
            | ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(643,axiom,
    tc_RealDef_Oreal @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(3154,plain,
    tc_RealDef_Oreal @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(751,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff__inverse2) ).

thf(3514,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(116,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Zero__neq__Suc) ).

thf(1560,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(9,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( C
          = ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__add__iff2) ).

thf(1274,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( C
            = ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( C
            = ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(496,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring) ).

thf(2673,plain,
    tc_Int_Oint @ class_Rings_Ocomm__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(51,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).

thf(1387,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(174,axiom,
    tc_Complex_Ocomplex @ class_Fields_Ofield,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Fields_Ofield) ).

thf(1723,plain,
    tc_Complex_Ocomplex @ class_Fields_Ofield,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(518,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__mono1) ).

thf(2733,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(34,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ c_Transcendental_Ocos @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re__rcis) ).

thf(1343,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ c_Transcendental_Ocos @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(120,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__inject__exp) ).

thf(1570,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( B = A )
           => ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) )
          & ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(323,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__n__not__n) ).

thf(2133,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(589,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__mult__pos2) ).

thf(2969,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(906,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__add__one) ).

thf(4091,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(1259,axiom,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__field,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__field) ).

thf(5366,plain,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__field,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1259]) ).

thf(918,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( C @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zmult2__eq) ).

thf(4128,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( C @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(666,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__0__less) ).

thf(3239,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(200,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__idom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__idom) ).

thf(1824,plain,
    tc_Int_Oint @ class_Rings_Olinordered__idom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(137,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) ).

thf(1619,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(330,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oone )
     => ( ( ( B @ c_Groups_Oone__class_Oone )
          = A )
      <=> ( A
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__reorient) ).

thf(2150,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oone )
     => ( ( ( A
            = ( B @ c_Groups_Oone__class_Oone ) )
         => ( ( B @ c_Groups_Oone__class_Oone )
            = A ) )
        & ( ( ( B @ c_Groups_Oone__class_Oone )
            = A )
         => ( A
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(562,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__less__cancel1) ).

thf(2890,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(415,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__add) ).

thf(2414,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(672,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__diff__assoc2) ).

thf(3250,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(244,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__commute) ).

thf(1938,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(422,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__add__self2) ).

thf(2433,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(114,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__Suc) ).

thf(1555,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(771,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__refl) ).

thf(3578,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(76,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).

thf(1457,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(855,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Oab__semigroup__idem__mult )
     => ( ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__idem) ).

thf(3894,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Oab__semigroup__idem__mult )
     => ( ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(475,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__idom )
     => ( ( C @ ( D @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( D @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__mult__less) ).

thf(2609,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__idom )
     => ( ( C @ ( D @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( D @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(932,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__left__mono) ).

thf(4187,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(153,axiom,
    tc_Nat_Onat @ class_Groups_Oordered__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__comm__monoid__add) ).

thf(1656,plain,
    tc_Nat_Onat @ class_Groups_Oordered__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(58,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__increasing__iff) ).

thf(1407,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(792,axiom,
    ! [A: $i,B: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__triangle__sub) ).

thf(3667,plain,
    ! [A: $i,B: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(583,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I8_J) ).

thf(2958,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(870,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__natfloor__le) ).

thf(3948,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(691,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) ).

thf(3307,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(648,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel1) ).

thf(3166,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(967,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__less__mono2) ).

thf(4295,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(1092,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_Odiff) ).

thf(4725,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).

thf(258,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__diff__1) ).

thf(1968,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(459,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oone) ).

thf(2565,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(655,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__add__less__zero__iff__single__add__less__zero) ).

thf(3198,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(689,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ? [C: $i] :
          ( A
          = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__Suc__ex__iff) ).

thf(3297,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(1103,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__le__mono) ).

thf(4757,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).

thf(619,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__strict__mono) ).

thf(3065,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(1199,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( ( B
          = ( A @ c_Nat_OSuc ) )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__Suc__eq) ).

thf(5177,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B
            = ( A @ c_Nat_OSuc ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B
            = ( A @ c_Nat_OSuc ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).

thf(229,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring__1) ).

thf(1888,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(806,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__lt__two__imp__zero__or__one) ).

thf(3718,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(996,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__fun__def) ).

thf(4402,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(206,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__eq__plus1__left) ).

thf(1836,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(999,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_n__less__m__mult__n) ).

thf(4417,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(608,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_SEQ_OBseq ) )
       => ? [C: $i] :
            ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_BseqD) ).

thf(3028,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_SEQ_OBseq ) )
       => ? [C: $i] :
            ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(377,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).

thf(2313,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(821,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__pow__pos) ).

thf(3763,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(809,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leD1) ).

thf(3727,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(10,axiom,
    ! [A: $i] :
      ( A
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_n__not__Suc__n) ).

thf(1281,plain,
    ! [A: $i] :
      ( A
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(314,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__semiring) ).

thf(2112,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(13,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__by__1) ).

thf(1291,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(719,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__1) ).

thf(3408,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(179,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__commute) ).

thf(1740,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(532,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__nonneg__nonneg) ).

thf(2769,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(889,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) ).

thf(4027,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(1231,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Ozero__right) ).

thf(5285,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1231]) ).

thf(811,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rabs__ratiotest__lemma) ).

thf(3731,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(108,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__real__of__nat) ).

thf(1541,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(552,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I4_J) ).

thf(2840,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C = A )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(1226,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_leD) ).

thf(5270,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1226]) ).

thf(848,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__add1__eq__le) ).

thf(3862,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(465,axiom,
    tc_Int_Oint @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__semigroup__add) ).

thf(2577,plain,
    tc_Int_Oint @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(90,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Transcendental_Osin )
      = ( B @ ( A @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( c_Complex_OIm @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__n__Im__cis__pow__n) ).

thf(1498,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Transcendental_Osin )
      = ( B @ ( A @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( c_Complex_OIm @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(225,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__Suc__right) ).

thf(1878,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(601,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__le__imp__le) ).

thf(3001,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(1120,axiom,
    tc_RealDef_Oreal @ class_Groups_Oordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(4808,plain,
    tc_RealDef_Oreal @ class_Groups_Oordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).

thf(354,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Transcendental_Osin )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__add) ).

thf(2245,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Transcendental_Osin )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(38,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ominus ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ominus) ).

thf(1350,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ominus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(1201,axiom,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__algebra,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__algebra) ).

thf(5189,plain,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__algebra,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).

thf(129,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring) ).

thf(1592,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(264,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__comm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__comm__semiring) ).

thf(1983,plain,
    tc_Nat_Onat @ class_Rings_Oordered__comm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(1164,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__eq__0__iff) ).

thf(4957,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1164]) ).

thf(1212,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_plus__nat_Oadd__0) ).

thf(5221,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1212]) ).

thf(674,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ? [B: $i] :
          ( A
          = ( B @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr0__conv__Suc) ).

thf(3254,plain,
    ! [A: $i] :
      ( ( ? [B: $i] :
            ( A
            = ( B @ c_Nat_OSuc ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ? [B: $i] :
            ( A
            = ( B @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(721,axiom,
    ( ( c_Complex_Oii @ ( c_Complex_ORe @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__i) ).

thf(3413,plain,
    ( ( c_Complex_Oii @ ( c_Complex_ORe @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(7,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Complex_Orcis ) @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Complex_Orcis ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DeMoivre2) ).

thf(1270,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Complex_Orcis ) @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_Complex_Orcis ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(185,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__one) ).

thf(1772,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(597,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__ge__zero) ).

thf(2990,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(1216,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__vector )
     => ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__ratiotest__lemma) ).

thf(5238,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__vector )
     => ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1216]) ).

thf(105,axiom,
    tc_Int_Oint @ class_Groups_Ocancel__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__comm__monoid__add) ).

thf(1534,plain,
    tc_Int_Oint @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(591,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq4) ).

thf(2973,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(396,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__class_Odistrib) ).

thf(2359,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(46,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__zmult__distrib2) ).

thf(1375,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(1113,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__nonpos__neg) ).

thf(4789,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).

thf(213,axiom,
    tc_Int_Oint @ class_Groups_Oordered__ab__group__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__group__add) ).

thf(1856,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(1005,axiom,
    tc_RealDef_Oreal @ class_Rings_Oring__1__no__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__1__no__zero__divisors) ).

thf(4429,plain,
    tc_RealDef_Oreal @ class_Rings_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).

thf(506,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__mult) ).

thf(2694,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(1030,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( B = A )
        | ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel__disj) ).

thf(4525,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B = A )
          | ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( B = A )
          | ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(442,axiom,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__add) ).

thf(2509,plain,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(950,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Oincseq ) )
      <=> ! [C: $i,D: $i] :
            ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_incseq__def) ).

thf(4233,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( ! [C: $i,D: $i] :
              ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ c_SEQ_Oincseq ) ) )
        & ( ( A @ ( B @ c_SEQ_Oincseq ) )
         => ! [C: $i,D: $i] :
              ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(624,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__antisym) ).

thf(3078,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(1144,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__refl) ).

thf(4883,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1144]) ).

thf(665,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_Nat__Transfer_Otsub ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) ).

thf(3237,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_Nat__Transfer_Otsub ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(869,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__right__cancel) ).

thf(3941,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( B = A )
         => ( ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(628,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr__implies__not0) ).

thf(3087,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(1124,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE__nat) ).

thf(4817,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).

thf(54,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) ).

thf(1397,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(970,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__conv2) ).

thf(4301,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(1139,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__strict__increasing2) ).

thf(4860,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).

thf(246,axiom,
    tc_Nat_Onat @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__add) ).

thf(1942,plain,
    tc_Nat_Onat @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(1180,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__right) ).

thf(5107,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).

thf(804,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add2) ).

thf(3714,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(1067,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__refl) ).

thf(4648,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(392,axiom,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__mult) ).

thf(2349,plain,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(312,axiom,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ c_Transcendental_Ocos ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re__cis) ).

thf(2108,plain,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ c_Transcendental_Ocos ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(1141,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B = A )
      <=> ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__eq__iff) ).

thf(4867,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1141]) ).

thf(230,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__less__add2) ).

thf(1889,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(62,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_SEQ_OBseq ) )
      <=> ? [C: $i] :
          ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Bseq__iff1a) ).

thf(1419,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_SEQ_OBseq ) ) )
        & ( ( A @ ( B @ c_SEQ_OBseq ) )
         => ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(873,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( C @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__mono) ).

thf(3954,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( C @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(895,axiom,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__gt__zero__imp__gt__one) ).

thf(4049,plain,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(30,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__Suc) ).

thf(1329,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(825,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_even__less__0__iff) ).

thf(3775,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(557,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__mono1) ).

thf(2877,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(1001,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( C @ c_NthRoot_Oroot ) )
          = ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ c_NthRoot_Oroot ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__commute) ).

thf(4421,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( C @ c_NthRoot_Oroot ) )
          = ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ c_NthRoot_Oroot ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(973,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__le__trans) ).

thf(4315,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(63,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__cancel1) ).

thf(1424,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B = A )
       => ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(729,axiom,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_RComplete_Onatceiling )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__neg) ).

thf(3441,plain,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_RComplete_Onatceiling )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(1220,axiom,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less0) ).

thf(5253,plain,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1220]) ).

thf(1248,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Oab__semigroup__idem__mult )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__idem) ).

thf(5337,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Lattices_Oab__semigroup__idem__mult )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1248]) ).

thf(1159,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_NthRoot_Oroot ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__pos2) ).

thf(4938,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_NthRoot_Oroot ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).

thf(1224,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_RComplete_Onatceiling )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__zero) ).

thf(5263,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_RComplete_Onatceiling )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1224]) ).

thf(60,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__1) ).

thf(1414,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(291,axiom,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra__1 )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_RealVector_Onorm__class_Onorm ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__one) ).

thf(2053,plain,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra__1 )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_RealVector_Onorm__class_Onorm ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(208,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Transcendental_Ocos )
      = ( B @ ( A @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( c_Complex_ORe @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__n__Re__cis__pow__n) ).

thf(1842,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Transcendental_Ocos )
      = ( B @ ( A @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( c_Complex_ORe @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(997,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__0__iff) ).

thf(4406,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( ( ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
            | ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
            | ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(934,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__add__le__zero__iff__single__add__le__zero) ).

thf(4191,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(124,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__div__algebra,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra) ).

thf(1584,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__div__algebra,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(1099,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__iff) ).

thf(4744,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).

thf(478,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) ).

thf(2613,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(609,axiom,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_Ononneg__bounded) ).

thf(3032,plain,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(593,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__less__eq) ).

thf(2978,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(486,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semidom) ).

thf(2649,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(260,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ c_Nat_OSuc )
         != A )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__lessI) ).

thf(1974,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ c_Nat_OSuc )
         != A )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(113,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__Suc) ).

thf(1552,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(1019,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__pos__neg) ).

thf(4482,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).

thf(276,axiom,
    ! [A: $i,B: $i] :
      ( ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__1__eq__mult__iff) ).

thf(2006,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) )
       => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__strict__increasing) ).

thf(1947,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(265,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Ozero__class_Ozero )
       != ( A @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__neq__one) ).

thf(1984,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Ozero__class_Ozero )
       != ( A @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(211,axiom,
    tc_Int_Oint @ class_Groups_Ogroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ogroup__add) ).

thf(1852,plain,
    tc_Int_Oint @ class_Groups_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(634,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__0) ).

thf(3115,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(171,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_Groups_Oabs__class_Oabs ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__idempotent) ).

thf(1715,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_Groups_Oabs__class_Oabs ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(708,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__leD) ).

thf(3364,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(802,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_NthRoot_Oroot ) )
            = ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__mult__lemma) ).

thf(3703,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_NthRoot_Oroot ) )
            = ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(201,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self1) ).

thf(1825,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(1170,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_q__pos__lemma) ).

thf(5063,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).

thf(1240,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__square) ).

thf(5317,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1240]) ).

thf(187,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__mult) ).

thf(1776,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(3,axiom,
    v_ga____ @ c_SEQ_Osubseq,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_g_I1_J) ).

thf(1263,plain,
    v_ga____ @ c_SEQ_Osubseq,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(789,axiom,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_Ononneg__bounded) ).

thf(3656,plain,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(788,axiom,
    ! [A: $i] : ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__natceiling) ).

thf(3654,plain,
    ! [A: $i] : ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(126,axiom,
    tc_Nat_Onat @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Opreorder) ).

thf(1588,plain,
    tc_Nat_Onat @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(451,axiom,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis )
      = ( A @ c_Transcendental_Osin @ ( A @ c_Transcendental_Ocos @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cis__def) ).

thf(2536,plain,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis )
      = ( A @ c_Transcendental_Osin @ ( A @ c_Transcendental_Ocos @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(500,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( C @ ( D @ c_Complex_Orcis ) @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Orcis ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rcis__mult) ).

thf(2681,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( C @ ( D @ c_Complex_Orcis ) @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Orcis ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(73,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult2__eq) ).

thf(1451,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(372,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__1) ).

thf(2304,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(830,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__le__power) ).

thf(3787,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(49,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Opreorder) ).

thf(1382,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(704,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__le__cancel__iff2) ).

thf(3354,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(203,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add) ).

thf(1831,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(467,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__less__eq) ).

thf(2581,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(435,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__add) ).

thf(2465,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(440,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ominus,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ominus) ).

thf(2505,plain,
    tc_Complex_Ocomplex @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(1129,axiom,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_odd__less__0) ).

thf(4829,plain,
    ! [A: $i] :
      ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).

thf(819,axiom,
    ! [A: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__Im__le__cmod) ).

thf(3760,plain,
    ! [A: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(1210,axiom,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__0__right) ).

thf(5217,plain,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1210]) ).

thf(290,axiom,
    tc_Int_Oint @ class_Rings_Oordered__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring) ).

thf(2052,plain,
    tc_Int_Oint @ class_Rings_Oordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(132,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Zero__not__Suc) ).

thf(1599,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(507,axiom,
    v_ga____ @ ( v_f____ @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( v_s @ ( tc_Nat_Onat @ ( tc_Complex_Ocomplex @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( c_Complex_OIm @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_g_I2_J) ).

thf(2697,plain,
    v_ga____ @ ( v_f____ @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( v_s @ ( tc_Nat_Onat @ ( tc_Complex_Ocomplex @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( c_Complex_OIm @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(618,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__idom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__idom) ).

thf(3064,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__idom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(173,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__inverse) ).

thf(1720,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(956,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__le) ).

thf(4259,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(325,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).

thf(2140,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(904,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__imp__less__left) ).

thf(4086,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(626,axiom,
    tc_RealDef_Oreal @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocomm__monoid__add) ).

thf(3083,plain,
    tc_RealDef_Oreal @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(646,axiom,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__le__one) ).

thf(3162,plain,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(945,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__le__add2) ).

thf(4223,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(140,axiom,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__natfloor__add__one__gt) ).

thf(1627,plain,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(400,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) ).

thf(2370,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(659,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__one__le__power) ).

thf(3215,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(986,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__less__eq__trans) ).

thf(4361,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(925,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_Oadd) ).

thf(4163,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(32,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( ( B
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_realpow__num__eq__if) ).

thf(1333,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( ( B
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(188,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oab__group__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__group__add) ).

thf(1779,plain,
    tc_Complex_Ocomplex @ class_Groups_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(1106,axiom,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero @ ( c_Complex_OIm @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_Ozero) ).

thf(4765,plain,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero @ ( c_Complex_OIm @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).

thf(301,axiom,
    tc_Complex_Ocomplex @ class_Rings_Odivision__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Odivision__ring) ).

thf(2076,plain,
    tc_Complex_Ocomplex @ class_Rings_Odivision__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(97,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring) ).

thf(1515,plain,
    tc_Int_Oint @ class_Rings_Olinordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(448,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ogroup__add) ).

thf(2529,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(214,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_realpow__minus__mult) ).

thf(1857,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(931,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add_Ocomm__neutral) ).

thf(4184,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(835,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
        = c_Complex_Oii )
    <=> ( ( A
          = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
        & ( B
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Complex__eq__i) ).

thf(3802,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
          = c_Complex_Oii ) )
      & ( ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
          = c_Complex_Oii )
       => ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(1191,axiom,
    tc_RealDef_Oreal @ class_Groups_Oab__group__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__group__add) ).

thf(5153,plain,
    tc_RealDef_Oreal @ class_Groups_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).

thf(195,axiom,
    tc_Int_Oint @ class_Groups_Ominus,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ominus) ).

thf(1814,plain,
    tc_Int_Oint @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(1066,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( D @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel__left__neg) ).

thf(4644,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( D @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(663,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__le__eq) ).

thf(3225,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(528,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( A @ c_Nat_OSuc @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__0__Suc) ).

thf(2761,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( A @ c_Nat_OSuc @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(935,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less) ).

thf(4195,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(1041,axiom,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__0__less__1) ).

thf(4567,plain,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).

thf(1111,axiom,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_Opos__bounded) ).

thf(4782,plain,
    ? [A: $i] :
      ( ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      & ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).

thf(941,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( F @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( E @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__neg__lemma) ).

thf(4212,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( F @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( E @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(774,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ! [D: $i] : ( B @ ( D @ ( A @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_SEQ_OBseq ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_BseqI) ).

thf(3587,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ! [D: $i] : ( B @ ( D @ ( A @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_SEQ_OBseq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(560,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( F @ ( E @ c_COMBK ) ) @ ( D @ ( E @ ( F @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
      = C ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_K__record__comp) ).

thf(2883,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( F @ ( E @ c_COMBK ) ) @ ( D @ ( E @ ( F @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(1017,axiom,
    ! [A: $i] :
      ~ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__real__of__nat__less__zero) ).

thf(4476,plain,
    ! [A: $i] :
      ~ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(170,axiom,
    ! [A: $i] :
      ( ( ( A @ c_Transcendental_Osin )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__zero__abs__cos__one) ).

thf(1712,plain,
    ! [A: $i] :
      ( ( ( A @ c_Transcendental_Osin )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(210,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__eq) ).

thf(1846,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(568,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc__eq__le) ).

thf(2915,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(425,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zpower__zadd__distrib) ).

thf(2440,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(1209,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel2) ).

thf(5209,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1209]) ).

thf(157,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
    <=> ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__power__eq__Suc__0__iff) ).

thf(1663,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
       => ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(485,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__iff1) ).

thf(2645,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(339,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__minus__commute) ).

thf(2189,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(724,axiom,
    tc_RealDef_Oreal @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__semigroup__add) ).

thf(3422,plain,
    tc_RealDef_Oreal @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(884,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ c_Nat_OSuc @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      <=> ( B
          = ( C @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__div__lemma) ).

thf(4004,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) )
         => ( ( B @ c_Nat_OSuc @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
        & ( ( ( B @ c_Nat_OSuc @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(271,axiom,
    tc_Int_Oint @ class_Rings_Oordered__ring__abs,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring__abs) ).

thf(1996,plain,
    tc_Int_Oint @ class_Rings_Oordered__ring__abs,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(236,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__eq__0) ).

thf(1906,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(150,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( c_Complex_OIm @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im) ).

thf(1648,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( c_Complex_OIm @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(790,axiom,
    tc_RealDef_Oreal @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Omonoid__add) ).

thf(3660,plain,
    tc_RealDef_Oreal @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(95,axiom,
    ! [A: $i,B: $i] :
      ~ ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__add__less1) ).

thf(1510,plain,
    ! [A: $i,B: $i] :
      ~ ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(414,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__comm__monoid__add) ).

thf(2413,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(919,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self__is__m) ).

thf(4131,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(1015,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__diff__triangle__ineq) ).

thf(4470,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).

thf(160,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__field,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__field) ).

thf(1680,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__field,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(1258,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        & ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel1) ).

thf(5359,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1258]) ).

thf(515,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__add__iff1) ).

thf(2717,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(741,axiom,
    tc_RealDef_Oreal @ class_Groups_Ominus,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ominus) ).

thf(3491,plain,
    tc_RealDef_Oreal @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(8,axiom,
    tc_Nat_Onat @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__semigroup__add) ).

thf(1273,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(576,axiom,
    tc_RealDef_Oreal @ class_Groups_Ozero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ozero) ).

thf(2941,plain,
    tc_RealDef_Oreal @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(685,axiom,
    tc_RealDef_Oreal @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ozero__neq__one) ).

thf(3289,plain,
    tc_RealDef_Oreal @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(679,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__le__dividend) ).

thf(3274,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(1217,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ? [C: $i] :
            ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      <=> ? [C: $i] :
          ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lemma__NBseq__def2) ).

thf(5241,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ? [C: $i] :
              ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) )
        & ( ? [C: $i] :
              ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
         => ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1217]) ).

thf(984,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__double__add__iff__zero__le__single__add) ).

thf(4354,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(458,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) )
      & ( ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__div__cancel__disj) ).

thf(2556,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) )
      & ( ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(915,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
           != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_no__zero__divisors) ).

thf(4116,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
           != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(705,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__apply) ).

thf(3358,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(282,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__cancel) ).

thf(2033,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(1186,axiom,
    tc_RealDef_Oreal @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Opreorder) ).

thf(5138,plain,
    tc_RealDef_Oreal @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).

thf(358,axiom,
    tc_Complex_Ocomplex @ class_Fields_Ofield__inverse__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Fields_Ofield__inverse__zero) ).

thf(2253,plain,
    tc_Complex_Ocomplex @ class_Fields_Ofield__inverse__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(851,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) ).

thf(3871,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(1071,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_RComplete_Onatceiling )
        = ( B @ ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__subtract) ).

thf(4657,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_RComplete_Onatceiling )
        = ( B @ ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(1167,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__imp__less__right) ).

thf(5048,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).

thf(1039,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__assoc) ).

thf(4562,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(675,axiom,
    ! [A: $i] :
      ( ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__zero__iff) ).

thf(3262,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(280,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__mono2) ).

thf(2028,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(982,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__real__def) ).

thf(4340,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(1025,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__increasing2) ).

thf(4503,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).

thf(832,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ c_NthRoot_Oroot ) @ ( A @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__strict__increasing) ).

thf(3792,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ c_NthRoot_Oroot ) @ ( A @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(563,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
              = ( C @ c_Groups_Ozero__class_Ozero ) )
          <=> ( ( A
                = ( C @ c_Groups_Ozero__class_Ozero ) )
              & ( B
                = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__nonneg__eq__0__iff) ).

thf(2894,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( ( ( A
                  = ( C @ c_Groups_Ozero__class_Ozero ) )
                & ( B
                  = ( C @ c_Groups_Ozero__class_Ozero ) ) )
             => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
                = ( C @ c_Groups_Ozero__class_Ozero ) ) )
            & ( ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
                = ( C @ c_Groups_Ozero__class_Ozero ) )
             => ( ( A
                  = ( C @ c_Groups_Ozero__class_Ozero ) )
                & ( B
                  = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(45,axiom,
    tc_Int_Oint @ class_Rings_Oordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__semiring) ).

thf(1374,plain,
    tc_Int_Oint @ class_Rings_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(987,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(4365,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(710,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__mono_H) ).

thf(3372,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(882,axiom,
    ! [A: $i] :
      ~ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__add__one__not__less__self) ).

thf(4000,plain,
    ! [A: $i] :
      ~ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(621,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__add )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(3070,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__add )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(287,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Oorder) ).

thf(2048,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(338,axiom,
    tc_Int_Oint @ class_Groups_Oone,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oone) ).

thf(2188,plain,
    tc_Int_Oint @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(734,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Ozero) ).

thf(3472,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(406,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oidom) ).

thf(2393,plain,
    tc_Complex_Ocomplex @ class_Rings_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(677,axiom,
    tc_RealDef_Oreal @ class_Rings_Oring__no__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__no__zero__divisors) ).

thf(3272,plain,
    tc_RealDef_Oreal @ class_Rings_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(56,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__add__abs) ).

thf(1403,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(409,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__0) ).

thf(2398,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(658,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__trans) ).

thf(3213,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(758,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__is__0__eq_H) ).

thf(3536,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(1112,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__assoc) ).

thf(4786,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).

thf(813,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__gt__0__iff) ).

thf(3741,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(761,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__left__mono) ).

thf(3547,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(205,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oring )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__diff__mult) ).

thf(1833,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oring )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(347,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( C @ class_Orderings_Oord )
        & ( D @ class_RealVector_Oreal__normed__vector ) )
     => ( ! [E: $i] :
            ( ( E @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( E @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ! [E: $i] :
            ( ( E @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B @ ( A @ hAPP ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lemmaCauchy) ).

thf(2225,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( C @ class_Orderings_Oord )
        & ( D @ class_RealVector_Oreal__normed__vector ) )
     => ( ! [E: $i] :
            ( ( E @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( E @ ( A @ hAPP ) @ ( B @ ( A @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ! [E: $i] :
            ( ( E @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B @ ( A @ hAPP ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( A @ hAPP ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(227,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__mult__self) ).

thf(1884,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( B @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(964,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__antisym__conv) ).

thf(4283,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( A = B )
           => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(136,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osimps_I3_J) ).

thf(1615,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(1171,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__antisym) ).

thf(5065,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1171]) ).

thf(924,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( B = A )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__eq__refl) ).

thf(4159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( B = A )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(625,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__pos__pos) ).

thf(3081,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(818,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_LIMSEQ__inverse__realpow__zero__lemma) ).

thf(3758,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(497,axiom,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) ).

thf(2674,plain,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(277,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__eq__plus1) ).

thf(2016,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(461,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__add__iff2) ).

thf(2568,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(1168,axiom,
    ! [A: $i] : ( v_r @ ( A @ ( v_s @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_assms) ).

thf(5050,plain,
    ! [A: $i] : ( v_r @ ( A @ ( v_s @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).

thf(106,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__commute) ).

thf(1535,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(779,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_monoseq__realpow) ).

thf(3604,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(991,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__trans) ).

thf(4381,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(1255,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__comm__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__mult__strict__left__mono) ).

thf(5350,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__comm__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1255]) ).

thf(849,axiom,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__vector,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__vector) ).

thf(3868,plain,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__vector,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(1054,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__ge__zero) ).

thf(4612,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(546,axiom,
    tc_RealDef_Oreal @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring__1) ).

thf(2817,plain,
    tc_RealDef_Oreal @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(1026,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__trans) ).

thf(4505,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(777,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__funD) ).

thf(3599,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(662,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__less__mono__lemma) ).

thf(3223,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(1032,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( D @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__left__neg) ).

thf(4537,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( D @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).

thf(615,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( C @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__mono) ).

thf(3051,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( C @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(1116,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self1__is__m) ).

thf(4796,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).

thf(23,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__1__mult) ).

thf(1312,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(31,axiom,
    tc_Nat_Onat @ class_Groups_Ominus,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ominus) ).

thf(1332,plain,
    tc_Nat_Onat @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(800,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__strict__mono) ).

thf(3700,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(234,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( D = C )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__eq) ).

thf(1897,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( ( B = A )
           => ( D = C ) )
          & ( ( D = C )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(80,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__assoc) ).

thf(1464,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(757,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__imp__zdiv__neg__iff) ).

thf(3532,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(606,axiom,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__sin__le__one) ).

thf(3023,plain,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(1037,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_realpow__Suc__le__self) ).

thf(4558,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(635,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__le__mult__iff) ).

thf(3118,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(817,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ c_Transcendental_Oln )
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__eq__zero__iff) ).

thf(3751,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
         => ( ( A @ c_Transcendental_Oln )
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ c_Transcendental_Oln )
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(908,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Ononneg__bounded) ).

thf(4096,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(382,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Opreorder) ).

thf(2326,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(797,axiom,
    ( ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( c_Complex_ORe @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__one) ).

thf(3695,plain,
    ( ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( c_Complex_ORe @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(521,axiom,
    tc_Int_Oint @ class_Groups_Oordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(2738,plain,
    tc_Int_Oint @ class_Groups_Oordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(1246,axiom,
    ( ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( c_Complex_OIm @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Im__one) ).

thf(5331,plain,
    ( ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( c_Complex_OIm @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1246]) ).

thf(1230,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Transcendental_Oln )
          = ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__mult) ).

thf(5282,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Transcendental_Oln )
          = ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1230]) ).

thf(218,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__idom) ).

thf(1865,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(1044,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__zpower__abs) ).

thf(4573,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(1177,axiom,
    tc_RealDef_Oreal @ class_Rings_Oidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oidom) ).

thf(5086,plain,
    tc_RealDef_Oreal @ class_Rings_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1177]) ).

thf(272,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__poly__code_I2_J) ).

thf(1997,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(1202,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__linear) ).

thf(5190,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).

thf(525,axiom,
    tc_Int_Oint @ class_Groups_Oordered__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__comm__monoid__add) ).

thf(2753,plain,
    tc_Int_Oint @ class_Groups_Oordered__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(828,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__gt__zero) ).

thf(3783,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(739,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__1__no__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
         != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_field__power__not__zero) ).

thf(3487,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__1__no__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
         != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(183,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
            = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ c_Nat_OSuc ) ) )
        & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__if) ).

thf(1765,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
            = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ c_Nat_OSuc ) ) )
        & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(782,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( B
          = ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__eq__add__iff2) ).

thf(3619,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( B
            = ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( B
            = ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(379,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__right__imp__eq) ).

thf(2319,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( C = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(1007,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ! [D: $i] :
                ( ( ( ( B @ ( D @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
                    = A )
                  & ( D @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
               => ( D = C ) )
            & ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = A )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_realpow__pos__nth__unique) ).

thf(4433,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ! [D: $i] :
                ( ( ( ( B @ ( D @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
                    = A )
                  & ( D @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
               => ( D = C ) )
            & ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = A )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(463,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__1) ).

thf(2573,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(329,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__less__imp__less__exp) ).

thf(2148,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(443,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( ( C = A )
          | ( D = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_crossproduct__eq) ).

thf(2510,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( ( C = A )
            | ( D = B ) )
         => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( ( C = A )
            | ( D = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(304,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
    <=> ( C = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__right__cancel) ).

thf(2079,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C = A )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( C = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(516,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__eq__if) ).

thf(2721,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(980,axiom,
    tc_RealDef_Oreal @ class_Rings_Ocomm__ring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__ring__1) ).

thf(4337,plain,
    tc_RealDef_Oreal @ class_Rings_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(1023,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__diff__ineq) ).

thf(4498,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).

thf(482,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) ).

thf(2624,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(595,axiom,
    tc_RealDef_Oreal @ class_Rings_Oordered__comm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__comm__semiring) ).

thf(2987,plain,
    tc_RealDef_Oreal @ class_Rings_Oordered__comm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(631,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__nonpos__nonpos) ).

thf(3106,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(1047,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( ( I @ ( J @ hAPP ) @ ( K @ hAPP ) )
        = ( F @ ( G @ hAPP ) @ ( H @ hAPP ) ) )
     => ( ( I @ ( J @ ( K @ ( C @ ( D @ ( E @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
        = ( F @ ( G @ ( H @ ( A @ ( D @ ( B @ c_Fun_Ocomp ) ) ) ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comp__cong) ).

thf(4586,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( ( I @ ( J @ hAPP ) @ ( K @ hAPP ) )
        = ( F @ ( G @ hAPP ) @ ( H @ hAPP ) ) )
     => ( ( I @ ( J @ ( K @ ( C @ ( D @ ( E @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
        = ( F @ ( G @ ( H @ ( A @ ( D @ ( B @ c_Fun_Ocomp ) ) ) ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).

thf(184,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__1) ).

thf(1771,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(912,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_NthRoot_Oroot ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__one) ).

thf(4107,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_NthRoot_Oroot ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(296,axiom,
    tc_Int_Oint @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__add) ).

thf(2065,plain,
    tc_Int_Oint @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(947,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__imp__le__right) ).

thf(4228,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(212,axiom,
    ( c_Complex_Oii
   != ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__i__not__one) ).

thf(1853,plain,
    ( c_Complex_Oii
   != ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(644,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__imp__zdiv__nonneg__iff) ).

thf(3155,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(348,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Osemiring )
     => ( ( A @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_combine__common__factor) ).

thf(2229,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Osemiring )
     => ( ( A @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(611,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__ge__zero) ).

thf(3038,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(971,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__nonpos__nonneg) ).

thf(4305,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(684,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      | ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__linear) ).

thf(3287,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      | ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(794,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__0__less__mult__iff) ).

thf(3683,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(101,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__left__commute) ).

thf(1524,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(826,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor @ ( A @ c_RComplete_Onatfloor @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__mult__natfloor) ).

thf(3779,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor @ ( A @ c_RComplete_Onatfloor @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(565,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__le__mono2) ).

thf(2906,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(1085,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__add) ).

thf(4699,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).

thf(37,axiom,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__semigroup__add) ).

thf(1349,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(676,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I7_J) ).

thf(3270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(367,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib2) ).

thf(2284,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(361,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__1__iff) ).

thf(2257,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(127,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__0) ).

thf(1589,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(88,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__Suc__right) ).

thf(1492,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(133,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oabs__class_Oabs ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          & ( C @ ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__diff__less__iff) ).

thf(1603,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__idom )
     => ( ( ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( C @ ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oabs__class_Oabs ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oabs__class_Oabs ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( C @ ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(1119,axiom,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Oarg @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__arg__i__mult__zero__neg) ).

thf(4805,plain,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Oarg @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).

thf(494,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) )
      = ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__Suc__le__raw) ).

thf(2669,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) )
      = ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(922,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__is__0) ).

thf(4139,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(654,axiom,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__algebra__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__algebra__1) ).

thf(3197,plain,
    tc_RealDef_Oreal @ class_RealVector_Oreal__normed__algebra__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(1253,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Divides_Otransfer__nat__int__function__closures_I1_J) ).

thf(5346,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1253]) ).

thf(901,axiom,
    ? [A: $i] :
    ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_Obounded) ).

thf(4073,plain,
    ? [A: $i] :
    ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(1075,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl) ).

thf(4667,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(928,axiom,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add) ).

thf(4176,plain,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(640,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel__left__pos) ).

thf(3138,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(661,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__imp__zdiv__neg__iff) ).

thf(3219,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(1006,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self2__is__id) ).

thf(4430,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(1033,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__conv) ).

thf(4541,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(638,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__add__strict) ).

thf(3133,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(321,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__ring) ).

thf(2129,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(899,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ c_NthRoot_Oroot ) )
          = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
      <=> ( A
          = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__eq__1__iff) ).

thf(4064,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
         => ( ( A @ ( B @ c_NthRoot_Oroot ) )
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) )
        & ( ( ( A @ ( B @ c_NthRoot_Oroot ) )
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
         => ( A
            = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(176,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Ocos )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__diff) ).

thf(1731,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_Transcendental_Ocos )
      = ( A @ c_Transcendental_Osin @ ( B @ c_Transcendental_Osin @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Transcendental_Ocos @ ( B @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(768,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__two) ).

thf(3572,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(930,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_NthRoot_Oroot ) )
            = ( A @ ( B @ c_NthRoot_Oroot ) @ ( C @ c_NthRoot_Oroot ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__pos__mult__exp) ).

thf(4181,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_NthRoot_Oroot ) )
            = ( A @ ( B @ c_NthRoot_Oroot ) @ ( C @ c_NthRoot_Oroot ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(977,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__neg) ).

thf(4332,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(472,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Oorder) ).

thf(2593,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(1069,axiom,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_RealVector_Onorm__class_Onorm ) )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__zero) ).

thf(4652,plain,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_RealVector_Onorm__class_Onorm ) )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(77,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring) ).

thf(1460,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(457,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zero) ).

thf(2553,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(48,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra) ).

thf(1381,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(433,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__trans__Suc) ).

thf(2461,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(251,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ono__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ono__zero__divisors) ).

thf(1952,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ono__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(1182,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__le__iff) ).

thf(5114,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).

thf(1260,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__neq) ).

thf(5367,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1260]) ).

thf(1207,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power_Opower_Opower__Suc) ).

thf(5202,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1207]) ).

thf(815,axiom,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__group__add__abs,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs) ).

thf(3747,plain,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__group__add__abs,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(1176,axiom,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra )
     => ? [B: $i] :
          ( ! [C: $i,D: $i] : ( B @ ( D @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Ononneg__bounded) ).

thf(5082,plain,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra )
     => ? [B: $i] :
          ( ! [C: $i,D: $i] : ( B @ ( D @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).

thf(1043,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__gt__zero__iff) ).

thf(4569,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).

thf(313,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__comm__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) ).

thf(2111,plain,
    tc_Int_Oint @ class_Rings_Olinordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(622,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__neg__pos) ).

thf(3073,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(537,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq) ).

thf(2780,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(362,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib) ).

thf(2267,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(1094,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero )
   != ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__zero__not__eq__one) ).

thf(4731,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero )
   != ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).

thf(427,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(2446,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(961,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos2) ).

thf(4277,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(527,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__neq__Zero) ).

thf(2757,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(738,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( E
         != ( F @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( B != A )
            & ( D = C ) )
         => ( ( B @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
           != ( A @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__scale__eq__noteq) ).

thf(3483,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( E
         != ( F @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( B != A )
            & ( D = C ) )
         => ( ( B @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
           != ( A @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(876,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Odecseq ) )
      <=> ! [C: $i,D: $i] :
            ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_decseq__def) ).

thf(3971,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( ! [C: $i,D: $i] :
              ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ c_SEQ_Odecseq ) ) )
        & ( ( A @ ( B @ c_SEQ_Odecseq ) )
         => ! [C: $i,D: $i] :
              ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(357,axiom,
    tc_Int_Oint @ class_Rings_Oring__1__no__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1__no__zero__divisors) ).

thf(2252,plain,
    tc_Int_Oint @ class_Rings_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(492,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__1__left) ).

thf(2664,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(512,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ge__natfloor__plus__one__imp__gt) ).

thf(2710,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(198,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__diff__inverse) ).

thf(1820,plain,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(466,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).

thf(2578,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(164,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) )
      = ( B @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__rcis) ).

thf(1688,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) )
      = ( B @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(70,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Oord ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Oord) ).

thf(1445,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Oord ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(148,axiom,
    ? [A: $i] :
      ( ( A @ ( v_s @ ( c_Complex_ORe @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_Fun_Ocomp ) ) ) ) @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) )
      & ( A @ c_SEQ_Osubseq ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact__096EX_Af_O_Asubseq_Af_A_G_Amonoseq_A_I_Fn_O_A_IRe_Ao_As_J_A_If_An_J_J_096) ).

thf(1643,plain,
    ? [A: $i] :
      ( ( A @ ( v_s @ ( c_Complex_ORe @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_Fun_Ocomp ) ) ) ) @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) )
      & ( A @ c_SEQ_Osubseq ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(861,axiom,
    ! [A: $i,B: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__triangle__ineq2) ).

thf(3917,plain,
    ! [A: $i,B: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(1218,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ! [C: $i,D: $i] :
            ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_monoI1) ).

thf(5248,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ! [C: $i,D: $i] :
            ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1218]) ).

thf(645,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__poly__0__left) ).

thf(3159,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(405,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__1) ).

thf(2390,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__less) ).

thf(3348,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(857,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__div__geq) ).

thf(3908,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(759,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__less__iff1) ).

thf(3539,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(300,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__vector,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__vector) ).

thf(2075,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__vector,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(1249,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ! [B: $i] :
        ? [C: $i] : ( A @ ( C @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reals__Archimedean3) ).

thf(5340,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ! [B: $i] :
        ? [C: $i] : ( A @ ( C @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1249]) ).

thf(1090,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_RComplete_Onatfloor @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__mono) ).

thf(4720,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_RComplete_Onatfloor @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).

thf(241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__diff__less) ).

thf(1926,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(586,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( C @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( D @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__quotient__lemma__neg) ).

thf(2962,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( C @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( D @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(949,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semidom) ).

thf(4232,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(393,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__Suc__0) ).

thf(2350,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(410,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__mult) ).

thf(2401,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(130,axiom,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ c_Transcendental_Osin ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im__cis) ).

thf(1594,plain,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ c_Transcendental_Osin ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(1205,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__le__imp__le__base) ).

thf(5197,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1205]) ).

thf(289,axiom,
    tc_Int_Oint @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ozero__neq__one) ).

thf(2051,plain,
    tc_Int_Oint @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(530,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
        ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Obounded) ).

thf(2765,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
        ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(725,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I1_J) ).

thf(3423,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( C = B )
       => ( ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(1128,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ~ ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__irrefl) ).

thf(4827,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Opreorder )
     => ~ ( A @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).

thf(573,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B != A )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__cases) ).

thf(2931,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B != A )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(667,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__refl) ).

thf(3241,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(541,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Oarg @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__arg__i__mult__zero__pos) ).

thf(2797,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Oarg @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(65,axiom,
    tc_Int_Oint @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__add) ).

thf(1435,plain,
    tc_Int_Oint @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(186,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__add) ).

thf(1775,plain,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(471,axiom,
    tc_Int_Oint @ class_Power_Opower,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Power_Opower) ).

thf(2592,plain,
    tc_Int_Oint @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(1153,axiom,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__cos__le__one) ).

thf(4918,plain,
    ! [A: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ c_Transcendental_Ocos @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).

thf(707,axiom,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J) ).

thf(3363,plain,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(692,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__left__cancel) ).

thf(3310,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( B = A )
         => ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(85,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__if) ).

thf(1477,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(803,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__real__square__gt__zero) ).

thf(3706,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
       => ~ ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ~ ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(641,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_NthRoot_Oroot ) )
        = ( A @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__power) ).

thf(3142,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_NthRoot_Oroot ) )
        = ( A @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(750,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add1) ).

thf(3512,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(650,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__cancel) ).

thf(3178,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(1022,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__ring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__ring__strict) ).

thf(4497,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__ring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).

thf(1059,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__nonneg__pos) ).

thf(4626,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(1244,axiom,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_RComplete_Onatfloor )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__neg) ).

thf(5326,plain,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_RComplete_Onatfloor )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1244]) ).

thf(1187,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = A )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_realpow__pos__nth) ).

thf(5139,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = A )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1187]) ).

thf(1147,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__le__one__le) ).

thf(4890,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).

thf(940,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat_Oadd__0__right) ).

thf(4209,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(594,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__poly__0__right) ).

thf(2984,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(193,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__1) ).

thf(1805,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(298,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__mult) ).

thf(2069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(555,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C
          = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__quotient__aux1) ).

thf(2867,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C
          = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(441,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__commute) ).

thf(2506,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(740,axiom,
    tc_RealDef_Oreal @ class_Rings_Oordered__cancel__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__cancel__semiring) ).

thf(3490,plain,
    tc_RealDef_Oreal @ class_Rings_Oordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(1003,axiom,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(4426,plain,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(730,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__diff__triangle__ineq) ).

thf(3444,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oabs__class_Oabs ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(1082,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__less__dividend) ).

thf(4690,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).

thf(867,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Oab__semigroup__idem__mult )
     => ( ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_times_Oidem) ).

thf(3936,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Lattices_Oab__semigroup__idem__mult )
     => ( ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(770,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__strict__left__mono) ).

thf(3576,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(81,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1__strict) ).

thf(1467,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(1011,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( D @ c_Groups_Ozero__class_Ozero @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          | ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__right__disj) ).

thf(4456,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( ( ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( D @ c_Groups_Ozero__class_Ozero @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
            | ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( D @ c_Groups_Ozero__class_Ozero @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
            | ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).

thf(235,axiom,
    tc_Int_Oint @ class_Groups_Ozero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ozero) ).

thf(1905,plain,
    tc_Int_Oint @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(284,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__0__left) ).

thf(2037,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(162,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__lessD) ).

thf(1684,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(350,axiom,
    ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero )
   != ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__0__neq__1) ).

thf(2234,plain,
    ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero )
   != ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(1143,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__less__iff) ).

thf(4879,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).

thf(966,axiom,
    ( c_Complex_Oii
   != ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__i__not__zero) ).

thf(4292,plain,
    ( c_Complex_Oii
   != ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(733,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ~ ( ? [D: $i] :
              ( ~ ( D @ ( C @ hAPP ) @ hBOOL )
              & ( B
                = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
          | ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__split__asm) ).

thf(3455,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( ? [D: $i] :
                ( ~ ( D @ ( C @ hAPP ) @ hBOOL )
                & ( B
                  = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
            | ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
              & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
       => ~ ( ? [D: $i] :
                ( ~ ( D @ ( C @ hAPP ) @ hBOOL )
                & ( B
                  = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
            | ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
              & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(1091,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatceiling )
        = ( A @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__add) ).

thf(4722,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_RComplete_Onatceiling )
        = ( A @ ( B @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).

thf(1122,axiom,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__nat__zero__code) ).

thf(4812,plain,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).

thf(843,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) @ c_RComplete_Onatfloor )
          = ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__div__nat) ).

thf(3849,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) @ c_RComplete_Onatfloor )
          = ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(156,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__ring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring__1) ).

thf(1662,plain,
    tc_Int_Oint @ class_Rings_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(1008,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__eq__less__or__eq) ).

thf(4441,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).

thf(921,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
        = ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__diff) ).

thf(4136,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
        = ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(308,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__abs) ).

thf(2096,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(223,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ozero) ).

thf(1873,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(726,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__one__le__iff__zero__less) ).

thf(3427,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(488,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).

thf(2652,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(687,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__one__le__zero) ).

thf(3293,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(929,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__eq) ).

thf(4177,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(571,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono1__neg) ).

thf(2927,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(501,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__Suc2) ).

thf(2684,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(342,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( C @ ( D @ c_Complex_Ocomplex_OComplex ) )
        = ( A @ ( B @ c_Complex_Ocomplex_OComplex ) ) )
    <=> ( ( C = A )
        & ( D = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex_Oinject) ).

thf(2202,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( C = A )
          & ( D = B ) )
       => ( ( C @ ( D @ c_Complex_Ocomplex_OComplex ) )
          = ( A @ ( B @ c_Complex_Ocomplex_OComplex ) ) ) )
      & ( ( ( C @ ( D @ c_Complex_Ocomplex_OComplex ) )
          = ( A @ ( B @ c_Complex_Ocomplex_OComplex ) ) )
       => ( ( C = A )
          & ( D = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(863,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I6_J) ).

thf(3921,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(33,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_left__add__mult__distrib) ).

thf(1340,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(1081,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( A @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__power__abs) ).

thf(4688,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( A @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).

thf(385,axiom,
    tc_Int_Oint @ class_Rings_Oidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oidom) ).

thf(2333,plain,
    tc_Int_Oint @ class_Rings_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(681,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__le) ).

thf(3278,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(474,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( B
         != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__zpower__abs__iff) ).

thf(2597,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(254,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__less__power__Suc) ).

thf(1958,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(577,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__less) ).

thf(2942,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(110,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(1547,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(514,axiom,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero @ c_Complex_Oexpi )
    = ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_expi__zero) ).

thf(2715,plain,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero @ c_Complex_Oexpi )
    = ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(98,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__self) ).

thf(1516,plain,
    ! [A: $i] :
      ( ( A
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(859,axiom,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le0) ).

thf(3913,plain,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(491,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Ocomplex_OComplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__add__def) ).

thf(2661,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Ocomplex_OComplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(340,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__diff__def) ).

thf(2192,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(326,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( A @ ( B @ ( E @ ( G @ ( C @ c_Fun_Ocomp ) ) ) ) @ ( D @ ( E @ ( F @ ( G @ c_Fun_Ocomp ) ) ) ) )
      = ( A @ ( B @ ( D @ ( C @ ( F @ ( G @ c_Fun_Ocomp ) ) ) ) @ ( E @ ( F @ ( C @ c_Fun_Ocomp ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__assoc) ).

thf(2143,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( A @ ( B @ ( E @ ( G @ ( C @ c_Fun_Ocomp ) ) ) ) @ ( D @ ( E @ ( F @ ( G @ c_Fun_Ocomp ) ) ) ) )
      = ( A @ ( B @ ( D @ ( C @ ( F @ ( G @ c_Fun_Ocomp ) ) ) ) @ ( E @ ( F @ ( C @ c_Fun_Ocomp ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(1197,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__inject__base) ).

thf(5171,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__cancel__right) ).

thf(3129,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(499,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Ocomm__neutral) ).

thf(2678,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(1065,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( B = A )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__less__linear) ).

thf(4641,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( B = A )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(781,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
     => ( ( ( C = B )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__cases) ).

thf(3608,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
     => ( ( ( C = B )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(1050,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__vector )
     => ( ! [E: $i] :
            ( ( E @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( E @ ( B @ hAPP ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( B @ ( D @ c_SEQ_OBseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_BseqI2_H) ).

thf(4593,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__vector )
     => ( ! [E: $i] :
            ( ( E @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( E @ ( B @ hAPP ) @ ( D @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( B @ ( D @ c_SEQ_OBseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(504,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors) ).

thf(2689,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(460,axiom,
    tc_Nat_Onat @ class_Power_Opower,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Power_Opower) ).

thf(2567,plain,
    tc_Nat_Onat @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(209,axiom,
    tc_Nat_Onat @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring__0) ).

thf(1845,plain,
    tc_Nat_Onat @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(1235,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__0) ).

thf(5294,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1235]) ).

thf(747,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq2__sym) ).

thf(3507,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(841,axiom,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__group__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__group__add) ).

thf(3842,plain,
    tc_RealDef_Oreal @ class_Groups_Oordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(633,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__pred) ).

thf(3112,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(180,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
    <=> ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
        | ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__is__1) ).

thf(1743,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
       => ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(55,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ c_RComplete_Onatceiling )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__real__of__nat) ).

thf(1400,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ c_RComplete_Onatceiling )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(1155,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__pow__pos2) ).

thf(4922,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).

thf(149,axiom,
    tc_Nat_Onat @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__mult) ).

thf(1647,plain,
    tc_Nat_Onat @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(245,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__add) ).

thf(1941,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(1175,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ? [C: $i] :
          ( A
          = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__iff__add) ).

thf(5074,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).

thf(189,axiom,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Divides_Osemiring__div) ).

thf(1780,plain,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(548,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl2) ).

thf(2821,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(657,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__le__eq__trans) ).

thf(3209,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(412,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__gt1) ).

thf(2407,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(1149,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__pos__neg2) ).

thf(4894,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).

thf(403,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( C @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_incr__lemma) ).

thf(2386,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( C @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Oabs__class_Oabs ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(604,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__asym) ).

thf(3019,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(670,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__mono2) ).

thf(3246,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(1029,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_COMBC ) ) ) ) @ hAPP ) )
      = ( B @ ( A @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBC__1) ).

thf(4522,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_COMBC ) ) ) ) @ hAPP ) )
      = ( B @ ( A @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).

thf(1215,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_Ocomplex__rec ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex_Orecs) ).

thf(5235,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) @ ( C @ ( D @ c_Complex_Ocomplex_Ocomplex__rec ) ) )
      = ( A @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1215]) ).

thf(786,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring__abs )
     => ( ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
       => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
          = ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__eq__mult) ).

thf(3642,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring__abs )
     => ( ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
       => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
          = ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(883,axiom,
    tc_RealDef_Oreal @ class_Power_Opower,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Power_Opower) ).

thf(4003,plain,
    tc_RealDef_Oreal @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(764,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__lt__0__iff) ).

thf(3554,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(700,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__decreasing) ).

thf(3344,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(656,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__norm__iff) ).

thf(3202,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ( A
           != ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A
           != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(735,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__div__less__self) ).

thf(3475,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(1,conjecture,
    v_r @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ ( v_r @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(1262,plain,
    ~ ( v_r @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(389,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__self) ).

thf(2337,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(380,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__apply) ).

thf(2322,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(11,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Oadd__right) ).

thf(1285,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(823,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__zless__mono2) ).

thf(3769,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(436,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__ring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring__strict) ).

thf(2468,plain,
    tc_Int_Oint @ class_Rings_Olinordered__ring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(266,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__0) ).

thf(1987,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(346,axiom,
    tc_Int_Oint @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(2224,plain,
    tc_Int_Oint @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(430,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add) ).

thf(2454,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(690,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__mono) ).

thf(3305,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(64,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__pred_H) ).

thf(1432,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(18,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Oadd) ).

thf(1300,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(796,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__not__le) ).

thf(3691,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(503,axiom,
    tc_Nat_Onat @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(2688,plain,
    tc_Nat_Onat @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(678,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__strict) ).

thf(3273,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(344,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) ).

thf(2219,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(262,axiom,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__mult) ).

thf(1980,plain,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(1192,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__cube) ).

thf(5154,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).

thf(26,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__not__Zero) ).

thf(1317,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(182,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ozero__neq__one) ).

thf(1763,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(107,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__diff__cancel) ).

thf(1538,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(509,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__Suc__shift) ).

thf(2701,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(952,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_RComplete_Onatfloor )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__zero) ).

thf(4241,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_RComplete_Onatfloor )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(564,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divisors__zero) ).

thf(2903,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(698,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ( B != A )
          & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__le) ).

thf(3329,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( ( ( B != A )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( B != A )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(649,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__eq__less__trans) ).

thf(3174,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(1096,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq2) ).

thf(4737,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).

thf(285,axiom,
    tc_Int_Oint @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Opreorder) ).

thf(2044,plain,
    tc_Int_Oint @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(438,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBC ) ) ) ) @ ( B @ ( tc_Nat_Onat @ ( C @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( C @ c_SEQ_OBseq ) )
       => ( B @ ( C @ c_SEQ_OBseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Bseq__offset) ).

thf(2500,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBC ) ) ) ) @ ( B @ ( tc_Nat_Onat @ ( C @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( C @ c_SEQ_OBseq ) )
       => ( B @ ( C @ c_SEQ_OBseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(1118,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__mult__pos) ).

thf(4802,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).

thf(192,axiom,
    ! [A: $i] :
      ( ( A @ ( c_Complex_OIm @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ c_Complex_Ocomplex_OComplex ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__surj) ).

thf(1802,plain,
    ! [A: $i] :
      ( ( A @ ( c_Complex_OIm @ hAPP ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ c_Complex_Ocomplex_OComplex ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(167,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add) ).

thf(1702,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(1058,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__le__lessD) ).

thf(4624,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(592,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_RComplete_Onatfloor )
        = ( B @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__subtract) ).

thf(2975,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ c_RComplete_Onatfloor )
        = ( B @ ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(1100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__le__not__le) ).

thf(4748,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).

thf(894,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__antisym) ).

thf(4046,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(92,axiom,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__add) ).

thf(1504,plain,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(125,axiom,
    ! [A: $i,B: $i] :
      ( ( B
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
     => ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__self__implies__10) ).

thf(1585,plain,
    ! [A: $i,B: $i] :
      ( ( B
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
     => ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(1084,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( D @ ( E @ ( F @ ( G @ ( H @ c_Fun_Ocomp ) ) ) ) )
        = ( A @ ( B @ ( F @ ( G @ ( C @ c_Fun_Ocomp ) ) ) ) ) )
     => ! [I: $i] :
          ( ( I @ ( D @ hAPP ) @ ( E @ hAPP ) )
          = ( I @ ( A @ hAPP ) @ ( B @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__eq__elim) ).

thf(4695,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( D @ ( E @ ( F @ ( G @ ( H @ c_Fun_Ocomp ) ) ) ) )
        = ( A @ ( B @ ( F @ ( G @ ( C @ c_Fun_Ocomp ) ) ) ) ) )
     => ! [I: $i] :
          ( ( I @ ( D @ hAPP ) @ ( E @ hAPP ) )
          = ( I @ ( A @ hAPP ) @ ( B @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).

thf(579,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__linear) ).

thf(2949,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(575,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__less) ).

thf(2939,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(226,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).

thf(1881,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(923,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
          | ( B
           != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__gt__zero__iff) ).

thf(4150,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( ( ( A
             != ( C @ c_Groups_Ozero__class_Ozero ) )
            | ( B
             != ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( A
             != ( C @ c_Groups_Ozero__class_Ozero ) )
            | ( B
             != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(998,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self1__is__id) ).

thf(4414,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(994,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I5_J) ).

thf(4392,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(943,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__not__less) ).

thf(4217,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(498,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Oexpi )
      = ( A @ c_Complex_Oexpi @ ( B @ c_Complex_Oexpi @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_expi__add) ).

thf(2675,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Oexpi )
      = ( A @ c_Complex_Oexpi @ ( B @ c_Complex_Oexpi @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(292,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add) ).

thf(2056,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(1241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra__1 )
     => ( A @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__power__ineq) ).

thf(5319,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra__1 )
     => ( A @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1241]) ).

thf(578,axiom,
    tc_RealDef_Oreal @ class_Rings_Oordered__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__ring) ).

thf(2948,plain,
    tc_RealDef_Oreal @ class_Rings_Oordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(798,axiom,
    tc_RealDef_Oreal @ class_Groups_Ocancel__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__comm__monoid__add) ).

thf(3697,plain,
    tc_RealDef_Oreal @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(898,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel1) ).

thf(4057,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( B = A )
         => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(787,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I9_J) ).

thf(3652,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(551,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__imp__less__right) ).

thf(2838,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(722,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Nat_OSuc @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq1) ).

thf(3415,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Nat_OSuc @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(598,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__le__1__iff) ).

thf(2992,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(263,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__ring__strict) ).

thf(1981,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(958,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__cancel__right) ).

thf(4267,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(281,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__inverse2) ).

thf(2030,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(946,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_NthRoot_Oroot ) )
        = ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__abs) ).

thf(4225,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( B @ c_NthRoot_Oroot ) )
        = ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(44,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__comm__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__comm__semiring) ).

thf(1372,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__comm__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(1131,axiom,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra )
     => ? [B: $i] :
          ( ! [C: $i,D: $i] : ( B @ ( D @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Opos__bounded) ).

thf(4838,plain,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra )
     => ? [B: $i] :
          ( ! [C: $i,D: $i] : ( B @ ( D @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( C @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).

thf(753,axiom,
    tc_RealDef_Oreal @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring__0) ).

thf(3520,plain,
    tc_RealDef_Oreal @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(709,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__gt__zero__cancel__iff) ).

thf(3366,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(727,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__comm__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__comm__semiring__strict) ).

thf(3433,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(207,axiom,
    ! [A: $i,B: $i] :
      ( ( B
        = ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor )
        = ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__power) ).

thf(1839,plain,
    ! [A: $i,B: $i] :
      ( ( B
        = ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_RComplete_Onatfloor )
        = ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(432,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) )
      = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__real__of__nat__cancel) ).

thf(2458,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) )
      = ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(1002,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__mono) ).

thf(4424,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(717,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__le__cancel__iff) ).

thf(3401,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ c_Transcendental_Oln @ ( B @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(584,axiom,
    tc_RealDef_Oreal @ class_Groups_Oordered__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add) ).

thf(2960,plain,
    tc_RealDef_Oreal @ class_Groups_Oordered__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(391,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__SucI) ).

thf(2347,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(1110,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__power) ).

thf(4780,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).

thf(808,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__imp__add1__zle) ).

thf(3725,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

thf(701,axiom,
    ! [A: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__Re__le__cmod) ).

thf(3346,plain,
    ! [A: $i] : ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(1079,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_Oadd) ).

thf(4678,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( c_Complex_ORe @ hAPP ) )
      = ( A @ ( c_Complex_ORe @ hAPP ) @ ( B @ ( c_Complex_ORe @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).

thf(1064,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__pos__nonneg) ).

thf(4639,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__comm__monoid__add )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).

thf(683,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__poly__code_I2_J) ).

thf(3284,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(1134,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__imp__less__left) ).

thf(4851,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).

thf(1053,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__real__less) ).

thf(4606,plain,
    ! [A: $i,B: $i] :
      ( ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).

thf(775,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_q__neg__lemma) ).

thf(3589,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(286,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) ).

thf(2045,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(629,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__iff__diff__le__0) ).

thf(3092,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(1161,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__left__pos) ).

thf(4945,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1161]) ).

thf(547,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B
          = ( A @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__SucE) ).

thf(2818,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B
          = ( A @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(620,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_RealVector_Oreal__normed__field )
        & ( C @ class_Fields_Ofield__inverse__zero ) )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__divide) ).

thf(3067,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_RealVector_Oreal__normed__field )
        & ( C @ class_Fields_Ofield__inverse__zero ) )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(1108,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neq0__conv) ).

thf(4770,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( A
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).

thf(257,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__add) ).

thf(1966,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(250,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__cancel2) ).

thf(1949,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(556,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__le__zero__iff) ).

thf(2870,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(550,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__no__zero__divisors )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__0__iff) ).

thf(2828,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__no__zero__divisors )
     => ( ( ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(27,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring) ).

thf(1321,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(237,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring__0) ).

thf(1913,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(303,axiom,
    tc_Int_Oint @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring__0) ).

thf(2078,plain,
    tc_Int_Oint @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(1000,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__not__sym) ).

thf(4419,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Opreorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(376,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__1) ).

thf(2310,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(852,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( B != A )
      <=> ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neq__iff) ).

thf(3874,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ( ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
         => ( B != A ) )
        & ( ( B != A )
         => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(476,axiom,
    tc_Int_Oint @ class_Groups_Oordered__ab__group__add__abs,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__group__add__abs) ).

thf(2611,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__group__add__abs,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(381,axiom,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__mult) ).

thf(2325,plain,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(694,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__one) ).

thf(3320,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(52,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring) ).

thf(1390,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(373,axiom,
    tc_Int_Oint @ class_Rings_Oring__no__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__no__zero__divisors) ).

thf(2305,plain,
    tc_Int_Oint @ class_Rings_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(490,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).

thf(2658,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(535,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__power) ).

thf(2775,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(177,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__mult__distrib) ).

thf(1734,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(1172,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__le__D1) ).

thf(5068,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1172]) ).

thf(470,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ozero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ozero) ).

thf(2591,plain,
    tc_Complex_Ocomplex @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(426,axiom,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis )
      = ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_Complex_Orcis ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cis__rcis__eq) ).

thf(2443,plain,
    ! [A: $i] :
      ( ( A @ c_Complex_Ocis )
      = ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_Complex_Orcis ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(511,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__nat_Odiff__0) ).

thf(2707,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(596,axiom,
    ! [A: $i] : ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__natceiling__ge) ).

thf(2988,plain,
    ! [A: $i] : ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(84,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__0) ).

thf(1476,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(1014,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__zero) ).

thf(4468,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).

thf(713,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__bin__lemma) ).

thf(3383,plain,
    ! [A: $i,B: $i] :
      ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(954,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_Odiff) ).

thf(4250,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ ( c_Complex_OIm @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(590,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__mono) ).

thf(2971,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(981,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq3) ).

thf(4338,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( A @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(1179,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_times__divide__eq__right) ).

thf(5104,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1179]) ).

thf(1126,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq2) ).

thf(4823,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).

thf(29,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ c_Transcendental_Osin @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im__rcis) ).

thf(1326,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Complex_Orcis ) @ ( c_Complex_OIm @ hAPP ) )
      = ( A @ c_Transcendental_Osin @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(810,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__leE) ).

thf(3729,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(917,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Omonoseq ) )
      <=> ( ! [C: $i,D: $i] :
              ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ! [C: $i,D: $i] :
              ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_monoseq__def) ).

thf(4121,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( ( ! [C: $i,D: $i] :
                ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
            | ! [C: $i,D: $i] :
                ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
         => ( A @ ( B @ c_SEQ_Omonoseq ) ) )
        & ( ( A @ ( B @ c_SEQ_Omonoseq ) )
         => ( ! [C: $i,D: $i] :
                ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( C @ ( A @ hAPP ) @ ( D @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
            | ! [C: $i,D: $i] :
                ( ( D @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( D @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(858,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__less__imp__less) ).

thf(3911,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(335,axiom,
    ! [A: $i,B: $i] :
      ( ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
    <=> ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
        | ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__is__add) ).

thf(2167,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) )
       => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(364,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( B = A )
      <=> ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__iff__diff__eq__0) ).

thf(2273,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(158,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__mult__distrib) ).

thf(1674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(831,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__0__right) ).

thf(3789,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(145,axiom,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__one) ).

thf(1639,plain,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(1227,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__diff__assoc) ).

thf(5272,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1227]) ).

thf(369,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__0) ).

thf(2290,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(356,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult) ).

thf(2251,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(839,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__le) ).

thf(3825,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(1080,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__diff__iff) ).

thf(4681,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( B = A )
           => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
              = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) )
          & ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
              = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).

thf(94,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring__1) ).

thf(1508,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(544,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Opos__bounded) ).

thf(2811,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
          ( ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(1114,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ~ ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__sum__squares__lt__zero) ).

thf(4791,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ~ ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).

thf(141,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__power) ).

thf(1629,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(275,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__less__SucD) ).

thf(2004,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(816,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat_Odiff__diff__eq) ).

thf(3748,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(1214,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero )
          = ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__zero__sym) ).

thf(5228,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ c_Groups_Ozero__class_Ozero )
            = ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ c_Groups_Ozero__class_Ozero )
            = ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1214]) ).

thf(784,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__le__cancel__iff1) ).

thf(3630,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( C @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(602,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
        = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) )
    <=> ( ( A
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
        & ( B
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Complex__eq__0) ).

thf(3003,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
          = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ c_Complex_Ocomplex_OComplex ) )
          = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(529,axiom,
    tc_Complex_Ocomplex @ class_Rings_Osemiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring) ).

thf(2764,plain,
    tc_Complex_Ocomplex @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(1178,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ( ! [D: $i] :
            ( ( B
              = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
           => ( D @ ( C @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__split) ).

thf(5087,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ! [D: $i] :
              ( ( B
                = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
             => ( D @ ( C @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
       => ( ! [D: $i] :
              ( ( B
                = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
             => ( D @ ( C @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).

thf(847,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Oarg @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__arg__i__mult__zero) ).

thf(3859,plain,
    ! [A: $i] :
      ( ( A
       != ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocomplex_OComplex ) @ c_Complex_Oarg @ c_Transcendental_Ocos )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(447,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).

thf(2526,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(416,axiom,
    tc_Nat_Onat @ class_Groups_Ocancel__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__comm__monoid__add) ).

thf(2416,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(115,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__comm__monoid__add) ).

thf(1558,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(574,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Fun_Ocomp ) ) ) ) )
        = B )
     => ( ( A @ ( C @ hAPP ) @ ( D @ hAPP ) )
        = ( A @ ( B @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__eq__dest__lhs) ).

thf(2934,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Fun_Ocomp ) ) ) ) )
        = B )
     => ( ( A @ ( C @ hAPP ) @ ( D @ hAPP ) )
        = ( A @ ( B @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(1009,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_NthRoot_Oroot ) )
          = ( A @ ( B @ c_NthRoot_Oroot ) @ ( C @ c_NthRoot_Oroot ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__mult__exp) ).

thf(4451,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_NthRoot_Oroot ) )
          = ( A @ ( B @ c_NthRoot_Oroot ) @ ( C @ c_NthRoot_Oroot ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).

thf(957,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__le__cancel1) ).

thf(4261,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(407,axiom,
    tc_Nat_Onat @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Olinorder) ).

thf(2394,plain,
    tc_Nat_Onat @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(216,axiom,
    ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_One__nat__def) ).

thf(1861,plain,
    ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(582,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono1) ).

thf(2956,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(878,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__eq__le__trans) ).

thf(3978,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(745,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__mult__iff) ).

thf(3497,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
            | ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
            | ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(1104,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__irrefl__nat) ).

thf(4759,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).

thf(68,axiom,
    ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__1) ).

thf(1440,plain,
    ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(21,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__Suc) ).

thf(1308,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) )
      = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(297,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ? [C: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( C @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reduce__poly__simple) ).

thf(2066,plain,
    ! [A: $i,B: $i] :
      ( ( B
       != ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ? [C: $i] : ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( C @ ( tc_Complex_Ocomplex @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(756,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__def) ).

thf(3529,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_Fun_Ocomp ) ) ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(1188,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__SucI) ).

thf(5144,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).

thf(318,axiom,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Osin @ ( A @ c_Transcendental_Ocos @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) )
      = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cmod__unit__one) ).

thf(2120,plain,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Osin @ ( A @ c_Transcendental_Ocos @ c_Complex_Ocomplex_OComplex ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) )
      = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(1184,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__left__mono) ).

thf(5128,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).

thf(481,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring) ).

thf(2623,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(418,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) ).

thf(2421,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(651,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leD2) ).

thf(3181,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(1136,axiom,
    ? [A: $i] :
    ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_Obounded) ).

thf(4855,plain,
    ? [A: $i] :
    ! [B: $i] : ( A @ ( B @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( c_Complex_OIm @ hAPP ) @ ( tc_RealDef_Oreal @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).

thf(1052,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( C @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = A )
         => ( ( A @ ( C @ c_NthRoot_Oroot ) )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__pos__unique) ).

thf(4602,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( C @ ( B @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = A )
         => ( ( A @ ( C @ c_NthRoot_Oroot ) )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(936,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Transcendental_Ocos )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cos__zero) ).

thf(4197,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Transcendental_Ocos )
    = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(696,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__neq__le__trans) ).

thf(3324,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(1165,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ! [D: $i] :
              ( ? [E: $i] :
                  ( ( B
                    = ( E @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                  & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
                  & ( E @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
             => ( D @ ( C @ hAPP ) @ hBOOL ) ) )
        & ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ! [D: $i] :
              ( ? [E: $i] :
                  ( ( B
                    = ( E @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                  & ( A @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
                  & ( E @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
             => ( D @ ( C @ hAPP ) @ hBOOL ) ) )
        & ( ( A
            = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__zdiv) ).

thf(4964,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( B
                      = ( E @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                    & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
                    & ( E @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
               => ( D @ ( C @ hAPP ) @ hBOOL ) ) )
          & ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( B
                      = ( E @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                    & ( A @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
                    & ( E @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
               => ( D @ ( C @ hAPP ) @ hBOOL ) ) )
          & ( ( A
              = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
           => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
       => ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( B
                      = ( E @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                    & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
                    & ( E @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
               => ( D @ ( C @ hAPP ) @ hBOOL ) ) )
          & ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( B
                      = ( E @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
                    & ( A @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
                    & ( E @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
               => ( D @ ( C @ hAPP ) @ hBOOL ) ) )
          & ( ( A
              = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
           => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1165]) ).

thf(567,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C
          = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__quotient__aux2) ).

thf(2912,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( C
          = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(773,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__imp__le__left) ).

thf(3585,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(1049,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__trans) ).

thf(4591,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(969,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__mono_H) ).

thf(4299,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(14,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__cancel__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__cancel__semiring) ).

thf(1294,plain,
    tc_Nat_Onat @ class_Rings_Oordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(1166,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ c_Nat__Transfer_Otsub ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ c_Nat__Transfer_Otsub ) )
          = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tsub__def) ).

thf(5040,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ c_Nat__Transfer_Otsub ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ c_Nat__Transfer_Otsub ) )
          = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1166]) ).

thf(397,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__mult) ).

thf(2362,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(333,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_RComplete_Onatfloor )
    = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__one) ).

thf(2162,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ c_RComplete_Onatfloor )
    = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(1174,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__add__one__self__le__self) ).

thf(5072,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).

thf(96,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Power_Opower ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Power_Opower) ).

thf(1513,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(232,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Complex_Ocis @ ( B @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Ocis ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cis__mult) ).

thf(1893,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Complex_Ocis @ ( B @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Complex_Ocis ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(283,axiom,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(2036,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(522,axiom,
    tc_Nat_Onat @ class_Groups_Oordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(2739,plain,
    tc_Nat_Onat @ class_Groups_Oordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(69,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__minus__commute) ).

thf(1442,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) )
        = ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oabs__class_Oabs ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(1204,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
     => ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( B @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__lemma) ).

thf(5194,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
     => ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
               => ( B @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1204]) ).

thf(763,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__vector )
     => ( ( C @ ( D @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__add__less) ).

thf(3552,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__vector )
     => ( ( C @ ( D @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(1157,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__eq2) ).

thf(4927,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1157]) ).

thf(888,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ c_NthRoot_Oroot ) @ ( A @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__increasing) ).

thf(4025,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ c_NthRoot_Oroot ) @ ( A @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(278,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ? [C: $i] :
          ( A
          = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__iff__Suc__add) ).

thf(2019,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(368,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).

thf(2287,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(1121,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__0__right) ).

thf(4809,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).

thf(468,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__zero) ).

thf(2587,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oordered__ab__group__add__abs )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(778,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__divide) ).

thf(3601,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(854,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__algebra )
     => ( ( C @ ( D @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__mult__less) ).

thf(3892,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__algebra )
     => ( ( C @ ( D @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(139,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__mono) ).

thf(1625,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(445,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring) ).

thf(2522,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(1257,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__imp__less__or__eq) ).

thf(5356,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( B = A )
          | ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1257]) ).

thf(580,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A != B )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I11_J) ).

thf(2951,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A != B )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(754,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__diff__iff) ).

thf(3521,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(1051,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ? [C: $i] :
          ( ( ( A @ c_Nat_OSuc @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = B )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_realpow__pos__nth2) ).

thf(4597,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ? [C: $i] :
          ( ( ( A @ c_Nat_OSuc @ ( C @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = B )
          & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).

thf(617,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__strict__decreasing) ).

thf(3062,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(647,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_n__less__n__mult__m) ).

thf(3164,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(319,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).

thf(2123,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(1200,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Transcendental_Osin )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sin__zero) ).

thf(5187,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Transcendental_Osin )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).

thf(893,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( B @ c_Transcendental_Oln )
            = ( A @ c_Transcendental_Oln ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__inj__iff) ).

thf(4039,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( B = A )
           => ( ( B @ c_Transcendental_Oln )
              = ( A @ c_Transcendental_Oln ) ) )
          & ( ( ( B @ c_Transcendental_Oln )
              = ( A @ c_Transcendental_Oln ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(36,axiom,
    tc_Complex_Ocomplex @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(1348,plain,
    tc_Complex_Ocomplex @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(147,axiom,
    tc_Nat_Onat @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oorder) ).

thf(1642,plain,
    tc_Nat_Onat @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(310,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Oincseq ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_incseq__imp__monoseq) ).

thf(2102,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Oincseq ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(885,axiom,
    tc_RealDef_Oreal @ class_Fields_Ofield,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Fields_Ofield) ).

thf(4014,plain,
    tc_RealDef_Oreal @ class_Fields_Ofield,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(122,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_NthRoot_Oroot ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__Suc__0) ).

thf(1578,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_NthRoot_Oroot ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(253,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ogroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ogroup__add) ).

thf(1957,plain,
    tc_Complex_Ocomplex @ class_Groups_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(914,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__right__mono) ).

thf(4114,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(1016,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__ge__0__iff) ).

thf(4472,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(402,axiom,
    tc_Int_Oint @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ono__zero__divisors) ).

thf(2385,plain,
    tc_Int_Oint @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(1087,axiom,
    tc_RealDef_Oreal @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ono__zero__divisors) ).

thf(4706,plain,
    tc_RealDef_Oreal @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).

thf(1076,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__ge__zero__iff) ).

thf(4670,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(829,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__neg__neg) ).

thf(3785,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring__strict )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(990,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__add__eq2) ).

thf(4378,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(360,axiom,
    ! [A: $i,B: $i] : ( B @ c_Nat_OSuc @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__Suc) ).

thf(2255,plain,
    ! [A: $i,B: $i] : ( B @ c_Nat_OSuc @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(545,axiom,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero @ ( c_Complex_ORe @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_Ozero) ).

thf(2815,plain,
    ( ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero @ ( c_Complex_ORe @ hAPP ) )
    = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(19,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__norm__cancel) ).

thf(1303,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Oabs__class_Oabs ) )
        = ( A @ ( B @ c_RealVector_Onorm__class_Onorm ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(102,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Odecseq ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_decseq__imp__monoseq) ).

thf(1527,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Odecseq ) )
       => ( A @ ( B @ c_SEQ_Omonoseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(699,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ? [C: $i] :
            ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      <=> ? [C: $i] :
          ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lemma__NBseq__def) ).

thf(3337,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ? [C: $i] :
              ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) )
        & ( ? [C: $i] :
              ( ! [D: $i] : ( C @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( C @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
         => ? [C: $i] :
            ! [D: $i] : ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( D @ ( A @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(1261,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__le__cancel1) ).

thf(5371,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1261]) ).

thf(288,axiom,
    tc_Int_Oint @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Divides_Osemiring__div) ).

thf(2050,plain,
    tc_Int_Oint @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(1145,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Orcis ) )
      = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rcis__zero__mod) ).

thf(4885,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Orcis ) )
      = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).

thf(795,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__1) ).

thf(3690,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(66,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__semidom )
     => ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__one) ).

thf(1436,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__semidom )
     => ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(890,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono2) ).

thf(4029,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(1062,axiom,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__nat_Osimps_I1_J) ).

thf(4631,plain,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(1004,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__strict__increasing) ).

thf(4427,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__comm__monoid__add )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(510,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Odiff__right) ).

thf(2704,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(417,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__strict__increasing__iff) ).

thf(2417,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(411,axiom,
    ! [A: $i] :
      ( ( ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( c_Complex_Oii @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( c_Complex_Oii @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unimodular__reduce__norm) ).

thf(2404,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) )
        = ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone ) )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( c_Complex_Oii @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( c_Complex_Oii @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Complex_Ocomplex @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(231,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring) ).

thf(1891,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(390,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_right__minus__eq) ).

thf(2340,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( B = A )
         => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(842,axiom,
    ! [A: $i] :
      ( ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__natfloor__eq__one) ).

thf(3843,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(879,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__le__nat) ).

thf(3982,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(642,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
      <=> ( ( A
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__zmult__eq__1__iff) ).

thf(3145,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( ( A
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
            & ( B
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) )
         => ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) )
        & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
         => ( ( A
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
            & ( B
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(542,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Ozero__left) ).

thf(2800,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(874,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ? [C: $i] :
            ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
            & ( B
              = ( C @ c_Nat_OSuc ) ) )
        | ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc__eq__0__disj) ).

thf(3956,plain,
    ! [A: $i,B: $i] :
      ( ( ( ? [C: $i] :
              ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( B
                = ( C @ c_Nat_OSuc ) ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ? [C: $i] :
              ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( B
                = ( C @ c_Nat_OSuc ) ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(856,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
    <=> ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__neq__iff) ).

thf(3897,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
       => ( B != A ) )
      & ( ( B != A )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(1040,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ c_SEQ_Osubseq )
     => ( A @ ( B @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_seq__suble) ).

thf(4565,plain,
    ! [A: $i,B: $i] :
      ( ( B @ c_SEQ_Osubseq )
     => ( A @ ( B @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(268,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__eq__less) ).

thf(1989,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(896,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
       => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__mult__neg__le) ).

thf(4051,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
       => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(41,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__mult__distrib) ).

thf(1363,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(24,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oone,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oone) ).

thf(1314,plain,
    tc_Complex_Ocomplex @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(1194,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I5_J) ).

thf(5159,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).

thf(983,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__iff1) ).

thf(4350,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(89,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Odiff__left) ).

thf(1495,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(1102,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__square) ).

thf(4755,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).

thf(154,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__one__right) ).

thf(1657,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(386,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ozero__neq__one) ).

thf(2334,plain,
    tc_Complex_Ocomplex @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(766,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
    <=> ( ( A
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
        & ( B
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__two__squares__add__zero__iff) ).

thf(3560,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_RealDef_Oreal @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(337,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__ring__abs ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__ring__abs) ).

thf(2186,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__ring__abs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(963,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I3_J) ).

thf(4281,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(765,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__decreasing) ).

thf(3558,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(1233,axiom,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__Suc) ).

thf(5290,plain,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1233]) ).

thf(805,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono1) ).

thf(3716,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(462,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring) ).

thf(2572,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(585,axiom,
    tc_RealDef_Oreal @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocomm__monoid__mult) ).

thf(2961,plain,
    tc_RealDef_Oreal @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(12,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__0) ).

thf(1288,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(169,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__comm__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict) ).

thf(1710,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__comm__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(480,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__iff2) ).

thf(2619,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(1152,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__or__eq__imp__le) ).

thf(4912,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).

thf(35,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__gt1__lemma) ).

thf(1346,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(135,axiom,
    ~ ! [A: $i] :
        ( ( A @ c_SEQ_Osubseq )
       => ~ ( A @ ( v_f____ @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( v_s @ ( tc_Nat_Onat @ ( tc_Complex_Ocomplex @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( c_Complex_OIm @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact__096_B_Bthesis_O_A_I_B_Bg_O_A_091_124_Asubseq_Ag_059_Amonoseq_A_I_Fn_O_AIm_A_Is_A_If_A_Ig_An_J_J_J_J_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096) ).

thf(1610,plain,
    ~ ! [A: $i] :
        ( ( A @ c_SEQ_Osubseq )
       => ~ ( A @ ( v_f____ @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( v_s @ ( tc_Nat_Onat @ ( tc_Complex_Ocomplex @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( c_Complex_OIm @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(1181,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ~ ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leE) ).

thf(5110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ~ ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).

thf(487,axiom,
    tc_Int_Oint @ class_Rings_Oring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring) ).

thf(2651,plain,
    tc_Int_Oint @ class_Rings_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(539,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv3) ).

thf(2783,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( ( A = B )
           => ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
          & ( ~ ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( A = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(1243,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__less__imp__less) ).

thf(5324,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1243]) ).

thf(975,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__eq__self__zero) ).

thf(4318,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(455,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Odiff) ).

thf(2549,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(261,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__commute) ).

thf(1977,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(607,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      | ( B = A )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__linear) ).

thf(3025,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      | ( B = A )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(47,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__self) ).

thf(1378,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(850,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq) ).

thf(3869,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add__abs )
     => ( A @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( B @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oabs__class_Oabs ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(112,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__strict) ).

thf(1550,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(995,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__less__Suc__eq) ).

thf(4395,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( A = B )
         => ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(900,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
        ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Obounded) ).

thf(4071,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ? [C: $i] :
        ! [D: $i] : ( C @ ( D @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(944,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__natfloor) ).

thf(4221,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(394,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Ocis )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Ocis @ ( A @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cis__real__of__nat__Suc__mult) ).

thf(2353,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Ocis )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Complex_Ocis @ ( A @ c_Complex_Ocis @ ( tc_Complex_Ocomplex @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(1228,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( A @ ( B @ c_SEQ_Omonoseq ) )
      <=> ( ! [C: $i] : ( C @ ( A @ hAPP ) @ ( C @ c_Nat_OSuc @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          | ! [C: $i] : ( C @ c_Nat_OSuc @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_monoseq__Suc) ).

thf(5275,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oorder )
     => ( ( ( ! [C: $i] : ( C @ ( A @ hAPP ) @ ( C @ c_Nat_OSuc @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ! [C: $i] : ( C @ c_Nat_OSuc @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ c_SEQ_Omonoseq ) ) )
        & ( ( A @ ( B @ c_SEQ_Omonoseq ) )
         => ( ! [C: $i] : ( C @ ( A @ hAPP ) @ ( C @ c_Nat_OSuc @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            | ! [C: $i] : ( C @ c_Nat_OSuc @ ( A @ hAPP ) @ ( C @ ( A @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1228]) ).

thf(439,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lemma__realpow__diff) ).

thf(2502,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( C @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(353,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osimps_I2_J) ).

thf(2241,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(1133,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_COMBK ) ) @ hAPP ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBK__1) ).

thf(4848,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( B @ ( C @ ( D @ c_COMBK ) ) @ hAPP ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).

thf(332,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__add__self1) ).

thf(2159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(39,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( B @ ( C @ c_SEQ_OBseq ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBC ) ) ) ) @ ( B @ ( tc_Nat_Onat @ ( C @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( C @ c_SEQ_OBseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Bseq__ignore__initial__segment) ).

thf(1352,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__vector )
     => ( ( B @ ( C @ c_SEQ_OBseq ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ ( tc_Nat_Onat @ c_COMBC ) ) ) ) @ ( B @ ( tc_Nat_Onat @ ( C @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( C @ c_SEQ_OBseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(588,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( C @ ( B @ hAPP ) )
          = ( C @ ( A @ hAPP ) ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ext) ).

thf(2966,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( C @ ( B @ hAPP ) )
          = ( C @ ( A @ hAPP ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(760,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__add__iff2) ).

thf(3543,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(6,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mono) ).

thf(1268,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(543,axiom,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__le__zero__cancel__iff) ).

thf(2803,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(104,axiom,
    ! [A: $i] : ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__natfloor__gt__diff__one) ).

thf(1532,plain,
    ! [A: $i] : ( A @ c_RComplete_Onatfloor @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(1117,axiom,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ c_NthRoot_Oroot ) )
      = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__zero) ).

thf(4799,plain,
    ! [A: $i] :
      ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( A @ c_NthRoot_Oroot ) )
      = ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).

thf(767,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__left__mono__neg) ).

thf(3570,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__ring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(50,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__antisym) ).

thf(1384,plain,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(720,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__le__imp__less) ).

thf(3411,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Olinordered__semiring__strict )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(1020,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I12_J) ).

thf(4484,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Oorder )
     => ( ( B != A )
       => ( ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).

thf(1229,axiom,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocis )
    = ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cis__zero) ).

thf(5280,plain,
    ( ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ c_Complex_Ocis )
    = ( tc_Complex_Ocomplex @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1229]) ).

thf(838,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__0__eq) ).

thf(3817,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(669,axiom,
    tc_RealDef_Oreal @ class_Groups_Oone,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oone) ).

thf(3245,plain,
    tc_RealDef_Oreal @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(939,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__n__not__le__n) ).

thf(4206,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(1089,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      <=> ( ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__eq__add__iff1) ).

thf(4713,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( ( ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = A )
         => ( ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
         => ( ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).

thf(868,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_RComplete_Onatfloor @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__natfloor) ).

thf(3939,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_RealDef_Oreal ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_RComplete_Onatfloor @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(452,axiom,
    tc_Int_Oint @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__mult) ).

thf(2539,plain,
    tc_Int_Oint @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(131,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1) ).

thf(1597,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(554,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__diff__le) ).

thf(2864,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(1163,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__le__cases) ).

thf(4955,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Orderings_Olinorder )
     => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).

thf(17,axiom,
    tc_Int_Oint @ class_Rings_Oordered__cancel__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__cancel__semiring) ).

thf(1299,plain,
    tc_Int_Oint @ class_Rings_Oordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(836,axiom,
    tc_RealDef_Oreal @ class_Groups_Ogroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ogroup__add) ).

thf(3812,plain,
    tc_RealDef_Oreal @ class_Groups_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(743,axiom,
    tc_RealDef_Oreal @ class_Rings_Olinordered__ring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__ring) ).

thf(3494,plain,
    tc_RealDef_Oreal @ class_Rings_Olinordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

thf(93,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).

thf(1505,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(959,axiom,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__zeroE) ).

thf(4271,plain,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(363,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Odiff) ).

thf(2270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(905,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr0I) ).

thf(4088,plain,
    ! [A: $i] :
      ( ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(1252,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__comm__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__mult__left__mono) ).

thf(5344,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__comm__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1252]) ).

thf(1077,axiom,
    tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( v_s @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact__0960_A_060_061_Acmod_A_Is_A0_J_096) ).

thf(4674,plain,
    tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( v_s @ hAPP ) @ ( tc_Complex_Ocomplex @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).

thf(603,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__le__eq__one) ).

thf(3013,plain,
    ! [A: $i] :
      ( ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_RComplete_Onatceiling @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(871,axiom,
    tc_RealDef_Oreal @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Oorder) ).

thf(3950,plain,
    tc_RealDef_Oreal @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(772,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
          & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__fun__def) ).

thf(3580,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
         => ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(294,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__power__div__base) ).

thf(2059,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(61,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Oprod__diff__prod) ).

thf(1416,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(933,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ~ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__square__less__zero) ).

thf(4189,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ~ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(72,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oord )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oord ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Oord) ).

thf(1449,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oord )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oord ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(807,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__gt__1__iff) ).

thf(3721,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(489,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).

thf(2655,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(526,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_poly__div__mult__right) ).

thf(2754,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(979,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2) ).

thf(4335,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(1127,axiom,
    tc_RealDef_Oreal @ class_Rings_Osemiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Osemiring) ).

thf(4826,plain,
    tc_RealDef_Oreal @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).

thf(1070,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__self) ).

thf(4655,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).

thf(495,axiom,
    tc_Int_Oint @ class_Groups_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) ).

thf(2672,plain,
    tc_Int_Oint @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(558,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__one__less__zero) ).

thf(2879,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(1028,axiom,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__ge__zero__imp__ge__one) ).

thf(4520,plain,
    ! [A: $i] :
      ( ( A @ c_Transcendental_Oln @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).

thf(960,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__add__eq1) ).

thf(4274,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ ( E @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ ( E @ ( D @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(152,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__Suc) ).

thf(1653,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(1086,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__mult__pos__le) ).

thf(4702,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
          | ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).

thf(1238,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add )
     => ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
      <=> ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__iff__diff__less__0) ).

thf(5307,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oordered__ab__group__add )
     => ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1238]) ).

thf(559,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__trans) ).

thf(2881,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Opreorder )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(783,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
      <=> ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__root__le__iff) ).

thf(3626,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ( ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( C @ c_NthRoot_Oroot ) @ ( B @ ( C @ c_NthRoot_Oroot ) @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( A @ ( B @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(1035,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__mono__neg) ).

thf(4555,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
         => ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).

thf(178,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__squared__diff__one__factored) ).

thf(1737,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(630,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
    <=> ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__power__nat__eq) ).

thf(3096,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(845,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__div__algebra )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( A @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_norm__power) ).

thf(3854,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__div__algebra )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_RealVector_Onorm__class_Onorm ) )
        = ( A @ ( B @ ( C @ c_RealVector_Onorm__class_Onorm ) @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(161,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) ).

thf(1681,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(1162,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
    <=> ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__diff1__eq) ).

thf(4949,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) )
      & ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).

thf(1107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__ab__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__imp__eq) ).

thf(4767,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__ab__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).

thf(718,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__poly__code_I1_J) ).

thf(3405,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(79,axiom,
    tc_Int_Oint @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Omult__zero) ).

thf(1463,plain,
    tc_Int_Oint @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(600,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_COMBB ) ) ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBB__1) ).

thf(2998,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( A @ ( B @ ( C @ ( D @ ( E @ ( F @ c_COMBB ) ) ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) @ ( C @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Oadd__left) ).

thf(1565,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(1223,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ ( tc_RealDef_Oreal @ c_SEQ_OBseq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Bseq__realpow) ).

thf(5261,plain,
    ! [A: $i] :
      ( ( A @ ( tc_RealDef_Oreal @ c_Groups_Ozero__class_Ozero @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( ( tc_RealDef_Oreal @ c_Groups_Oone__class_Oone @ ( A @ ( tc_RealDef_Oreal @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
       => ( A @ ( tc_RealDef_Oreal @ c_Power_Opower__class_Opower @ hAPP ) @ ( tc_RealDef_Oreal @ c_SEQ_OBseq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1223]) ).

thf(1034,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero )
          = A )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__reorient) ).

thf(4547,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ c_Groups_Ozero__class_Ozero )
            = A ) )
        & ( ( ( B @ c_Groups_Ozero__class_Ozero )
            = A )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).

thf(423,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ono__zero__divisors) ).

thf(2436,plain,
    tc_Complex_Ocomplex @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(352,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diffs0__imp__equal) ).

thf(2238,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(1098,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__le__add1) ).

thf(4742,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).

thf(305,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__geq) ).

thf(2087,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
     => ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(866,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__0__left) ).

thf(3933,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(1138,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__less__Suc) ).

thf(4858,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1138]) ).

thf(82,axiom,
    v_f____ @ ( v_s @ ( tc_Nat_Onat @ ( tc_Complex_Ocomplex @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( c_Complex_ORe @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_f_I2_J) ).

thf(1468,plain,
    v_f____ @ ( v_s @ ( tc_Nat_Onat @ ( tc_Complex_Ocomplex @ ( tc_Nat_Onat @ c_COMBB ) ) ) ) @ ( c_Complex_ORe @ ( tc_Nat_Onat @ ( tc_RealDef_Oreal @ ( tc_Complex_Ocomplex @ c_COMBB ) ) ) ) @ ( tc_RealDef_Oreal @ c_SEQ_Omonoseq ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(221,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add) ).

thf(1870,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(533,axiom,
    tc_RealDef_Oreal @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring) ).

thf(2771,plain,
    tc_RealDef_Oreal @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(793,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
     => ( ( ( ~ ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( A @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) )
          & ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) )
       => ( ( C
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( C @ ( D @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__div__eq) ).

thf(3669,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
     => ( ( ( ~ ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL )
              & ( A @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL ) ) )
          & ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
           => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) @ hAPP ) @ hBOOL )
              & ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) @ hAPP ) @ hBOOL ) ) ) )
       => ( ( C
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( C @ ( D @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(1095,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) ).

thf(4734,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).

thf(8635,plain,
    $false,
    inference(e,[status(thm)],[3021,1411,1633,3798,4022,3434,2331,4728,1549,2280,1501,4201,4560,1718,2612,4094,1469,1369,4835,1823,2306,4464,3053,4101,2336,2131,1454,2072,5329,3661,3399,1995,3930,4169,2921,4290,3477,3698,2463,3397,1315,2595,2427,3969,5205,4015,4877,1818,1295,1686,4166,1924,4075,3762,1808,3509,2248,1544,3108,2114,1692,5348,4487,4479,2077,4931,2309,4376,2117,1406,2104,3317,5168,4500,3813,5292,2395,2908,2062,3549,3377,3040,2590,5297,2099,5224,1591,5265,4215,1971,4614,1486,4105,2195,2686,4575,3135,2740,5288,2448,4279,3781,5192,4762,3276,5120,3217,3794,1608,4633,3049,3745,4739,4110,1867,2184,2316,3446,5146,1872,2712,4650,2627,3253,1988,5161,3481,1954,2844,1354,2146,4941,2216,3350,1391,4618,1935,1921,2886,1724,1366,1265,2531,4386,3634,5252,3766,3994,3915,2548,1782,2825,1529,4665,4322,5352,5342,1887,3195,2735,2667,4692,3495,3574,1323,4778,1832,2126,2356,4243,3248,4238,3976,2328,5306,4896,1792,2698,1760,1660,1298,1860,1264,4334,3125,5150,5130,1577,1455,3280,1992,4568,1462,3043,2691,4862,5199,1704,1568,1963,4253,4119,3517,3374,5256,4358,2552,4664,2767,2373,4622,3919,4317,3084,2929,4675,2635,5052,3734,4368,2520,2730,3951,2158,1641,3835,4383,2301,1815,1875,4793,4507,2388,2533,2456,2222,3322,2027,3075,2090,3702,3525,4815,5156,2777,5259,3771,1489,2237,2616,1946,4857,2996,4454,3295,2782,1632,4297,2469,5333,4820,3453,2451,1828,2925,2291,4134,2254,1677,2540,5311,3290,3923,1960,1581,2335,2058,4660,1896,1447,3505,2772,3327,2750,1943,4707,4892,4628,3492,3184,3389,2964,2367,2437,1928,4055,4230,4033,2281,4842,5220,2523,3036,1622,2424,5343,2137,5321,4307,3361,2002,4536,3243,3606,1521,3510,1564,1863,5174,3857,4199,3591,4753,1691,3984,2118,5070,4557,4910,1868,2954,1651,4853,3229,4589,4629,2574,4888,1438,3852,1306,2364,2164,1915,2093,4076,2737,1311,2232,4576,4031,4100,2790,2000,1636,3418,4925,2409,4466,2428,1519,4920,2147,3884,3154,3514,1560,1274,2673,1387,1723,2733,1343,1570,2133,2969,4091,5366,4128,3239,1824,1619,2150,2890,2414,3250,1938,2433,1555,3578,1457,3894,2609,4187,1656,1407,3667,2958,3948,3307,3166,4295,4725,1968,2565,3198,3297,4757,3065,5177,1888,3718,4402,1836,4417,3028,2313,3763,3727,1281,2112,1291,3408,1740,2769,4027,5285,3731,1541,2840,5270,3862,2577,1498,1878,3001,4808,2245,1350,5189,1592,1983,4957,5221,3254,3413,1270,1772,2990,5238,1534,2973,2359,1375,4789,1856,4429,2694,4525,2509,4233,3078,4883,3237,3941,3087,4817,1397,4301,4860,1942,5107,3714,4648,2349,2108,4867,1889,1419,3954,4049,1329,3775,2877,4421,4315,1424,3441,5253,5337,4938,5263,1414,2053,1842,4406,4191,1584,4744,2613,3032,2978,2649,1974,1552,4482,2006,1947,1984,1852,3115,1715,3364,3703,1825,5063,5317,1776,1263,3656,3654,1588,2536,2681,1451,2304,3787,1382,3354,1831,2581,2465,2505,4829,3760,5217,2052,1599,2697,3064,1720,4259,2140,4086,3083,3162,4223,1627,2370,3215,4361,4163,1333,1779,4765,2076,1515,2529,1857,4184,3802,5153,1814,4644,3225,2761,4195,4567,4782,4212,3587,2883,4476,1712,1846,2915,2440,5209,1663,2645,2189,3422,4004,1996,1906,1648,3660,1510,2413,4131,4470,1680,5359,2717,3491,1273,2941,3289,3274,5241,4354,2556,4116,3358,2033,5138,2253,3871,4657,5048,4562,3262,2028,4340,4503,3792,2894,1374,4365,3372,4000,3070,2048,2188,3472,2393,3272,1403,2398,3213,3536,4786,3741,3547,1833,2225,1884,4283,1615,5065,4159,3081,3758,2674,2016,2568,5050,1535,3604,4381,5350,3868,4612,2817,4505,3599,3223,4537,3051,4796,1312,1332,3700,1897,1464,3532,3023,4558,3118,3751,4096,2326,3695,2738,5331,5282,1865,4573,5086,1997,5190,2753,3783,3487,1765,3619,2319,4433,2573,2148,2510,2079,2721,4337,4498,2624,2987,3106,4586,1771,4107,2065,4228,1853,3155,2229,3038,4305,3287,3683,1524,3779,2906,4699,1349,3270,2284,2257,1589,1492,1603,4805,2669,4139,3197,5346,4073,4667,4176,3138,3219,4430,4541,3133,2129,4064,1731,3572,4181,4332,2593,4652,1460,2553,1381,2461,1952,5114,5367,5202,3747,5082,4569,2111,3073,2780,2267,4731,2446,4277,2757,3483,3971,2252,2664,2710,1820,2578,1688,1445,1643,3917,5248,3159,2390,3348,3908,3539,2075,5340,4720,1926,2962,4232,2350,2401,1594,5197,2051,2765,3423,4827,2931,3241,2797,1435,1775,2592,4918,3363,3310,1477,3706,3142,3512,3178,4497,4626,5326,5139,4890,4209,2984,1805,2069,2867,2506,3490,4426,3444,4690,3936,3576,1467,4456,1905,2037,1684,2234,4879,4292,3455,4722,4812,3849,1662,4441,4136,2096,1873,3427,2652,3293,4177,2927,2684,2202,3921,1340,4688,2333,3278,2597,1958,2942,1547,2715,1516,3913,2661,2192,2143,5171,3129,2678,4641,3608,4593,2689,2567,1845,5294,3507,3842,3112,1743,1400,4922,1647,1941,5074,1780,2821,3209,2407,4894,2386,3019,3246,4522,5235,3642,4003,3554,3344,3202,3475,1262,2337,2322,1285,3769,2468,1987,2224,2454,3305,1432,1300,3691,2688,3273,2219,1980,5154,1317,1763,1538,2701,4241,2903,3329,3174,4737,2044,2500,4802,1802,1702,4624,2975,4748,4046,1504,1585,4695,2949,2939,1881,4150,4414,4392,4217,2675,2056,5319,2948,3697,4057,3652,2838,3415,2992,1981,4267,2030,4225,1372,4838,3520,3366,3433,1839,2458,4424,3401,2960,2347,4780,3725,3346,4678,4639,3284,4851,4606,3589,2045,3092,4945,2818,3067,4770,1966,1949,2870,2828,1321,1913,2078,4419,2310,3874,2611,2325,3320,1390,2305,2658,2775,1734,5068,2591,2443,2707,2988,1476,4468,3383,4250,2971,4338,5104,4823,1326,3729,4121,3911,2167,2273,1674,3789,1639,5272,2290,2251,3825,4681,1508,2811,4791,1629,2004,3748,5228,3630,3003,2764,5087,3859,2526,2416,1558,2934,4451,4261,2394,1861,2956,3978,3497,4759,1440,1308,2066,3529,5144,2120,5128,2623,2421,3181,4855,4602,4197,3324,4964,2912,3585,4591,4299,1294,5040,2362,2162,5072,1513,1893,2036,2739,1442,5194,3552,4927,4025,2019,2287,4809,2587,3601,3892,1625,2522,5356,2951,3521,4597,3062,3164,2123,5187,4039,1348,1642,2102,4014,1578,1957,4114,4472,2385,4706,4670,3785,4378,2255,2815,1303,1527,3337,5371,2050,4885,3690,1436,4029,4631,4427,2704,2417,2404,1891,2340,3843,3982,3145,2800,3956,3897,4565,1989,4051,1363,1314,5159,4350,1495,4755,1657,2334,3560,2186,4281,3558,5290,3716,2572,2961,1288,1710,2619,4912,1346,1610,5110,2651,2783,5324,4318,2549,1977,3025,1378,3869,1550,4395,4071,4221,2353,5275,2502,2241,4848,2159,1352,2966,3543,1268,2803,1532,4799,3570,1384,3411,4484,5280,3817,3245,4206,4713,3939,2539,1597,2864,4955,1299,3812,3494,1505,4271,2270,4088,5344,4674,3013,3950,3580,2059,1416,4189,1449,3721,2655,2754,4335,4826,4655,2672,2879,4520,4274,1653,4702,5307,2881,3626,4555,1737,3096,3854,1681,4949,4767,3405,1463,2998,1565,5261,4547,2436,2238,4742,2087,3933,4858,1468,1870,2771,3669,4734]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : SWW204+1 : TPTP v9.3.1. Released v5.2.0.
% 0.00/0.08  % Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.16/15.44  % Computer : n019.cluster.edu
% 0.16/15.44  % Model    : x86_64 x86_64
% 0.16/15.44  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/15.44  % Memory   : 8046.5625MB
% 0.16/15.44  % OS       : Linux 6.8.0-71-generic
% 0.16/15.44  % CPULimit : 300
% 0.16/15.44  % WCLimit  : 300
% 0.16/15.44  % DateTime : Sat Sep 26 15:28:21 UTC 2026
% 0.16/15.44  % CPUTime  : 
% 0.16/15.44  Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.99/16.03  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 2.54/16.55  % [INFO] 	 Parsing done (516ms). 
% 2.54/16.57  % [INFO] 	 Running in sequential loop mode. 
% 3.49/16.95  % [INFO] 	 eprover registered as external prover. 
% 3.49/16.96  % [INFO] 	 Scanning for conjecture ... 
% 4.40/17.29  % [INFO] 	 Found a conjecture (or negated_conjecture) and 1263 axioms. Running axiom selection ... 
% 6.17/17.66  % [INFO] 	 Axiom selection finished. Selected 1259 axioms (removed 4 axioms). 
% 8.30/18.29  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 8.66/18.36  % [INFO] 	 Type checking passed. 
% 8.66/18.36  % [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 ... 
% 62.42/32.48  % External prover 'e' found a proof!
% 62.42/32.48  % [INFO] 	 Killing All external provers ... 
% 62.42/32.48  % Time passed: 16898ms (effective reasoning time: 15904ms)
% 62.42/32.48  % Solved by 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)>
% 62.42/32.50  % Axioms used in derivation (1259): fact_xt1_I1_J, fact_monoseq__def, arity_fun__Orderings_Oorder, fact_le__imp__diff__is__add, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J, fact_power__abs, fact_Re__cis, fact_natfloor__power, fact_zle__refl, fact_add__le__less__mono, fact_mult__strict__left__mono__neg, fact_order__le__neq__trans, fact_Nat_Oadd__0__right, fact_real__squared__diff__one__factored, fact_div__mult__mult1, fact_le__SucI, arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra, arity_Polynomial__Opoly__Rings_Osemiring__0, fact_abs__of__pos, fact_real__le__refl, arity_Complex__Ocomplex__Rings_Oidom, fact_nat_Osimps_I3_J, fact_real__less__def, arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_mult__right__le__imp__le, arity_Nat__Onat__Groups_Ocancel__semigroup__add, fact_real__le__trans, fact_power__strict__decreasing, arity_Complex__Ocomplex__Groups_Omonoid__add, arity_Int__Oint__Rings_Ocomm__semiring, fact_natfloor__real__of__nat, fact_pos__add__strict, arity_Int__Oint__Rings_Osemiring, fact_mult__poly__0__right, arity_RealDef__Oreal__Orderings_Oord, fact_add__strict__right__mono, arity_Int__Oint__Orderings_Olinorder, fact_abs__sin__le__one, fact_Re_Oadd, fact_natfloor__add, fact_mult__1__left, fact_diff__0__eq__0, fact_zdiv__eq__0__iff, fact_diff__add__inverse, fact_abs__minus__commute, fact_xt1_I4_J, arity_RealDef__Oreal__Groups_Ozero, fact_trans__less__add1, fact_mult__nonneg__nonpos, fact_q__pos__lemma, fact_add__poly__code_I2_J, fact_abs__ge__self, arity_RealDef__Oreal__Rings_Oordered__comm__semiring, arity_Complex__Ocomplex__Groups_Oab__group__add, fact_diff__0__right, fact_mult__less__cancel__left__disj, fact_nat__add__left__commute, fact_real__mult__left__cancel, fact_mult__nonpos__nonpos, fact_order__le__less__trans, fact_abs__Re__le__cmod, fact_real__root__less__iff, arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors, fact_power__power__power, fact_real__of__nat__zero, fact_power__inject__base, fact_norm__power, fact_sin__le__one, fact_le__eq__less__or__eq, arity_Nat__Onat__Rings_Ono__zero__divisors, fact_nat__less__cases, fact_one__is__add, fact_diff__le__mono2, arity_Int__Oint__Rings_Olinordered__semiring__strict, arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add, fact_Complex__eq__1, fact_real__of__nat__diff, fact_termination__basic__simps_I5_J, arity_Polynomial__Opoly__Rings_Ocomm__semiring__0, fact_norm__mult__less, fact_odd__nonzero, fact_mult_Opos__bounded, fact_norm__diff__triangle__ineq, fact_Im_Opos__bounded, fact_mult__eq__0__iff, fact_Re_Ononneg__bounded, fact_less__imp__le__nat, fact_combine__common__factor, fact_le__imp__0__less, fact_add__leD1, fact__0960_A_060_061_Acmod_A_Is_A0_J_096, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J, fact_add__less__mono, fact_mult__0, fact_cis__def, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J, fact_real__root__one, arity_RealDef__Oreal__Groups_Ominus, arity_Nat__Onat__Orderings_Oorder, fact_mult__right__less__imp__less, fact_monoI1, fact_complex__zero__def, fact_abs__eq__mult, fact_split__mult__neg__le, fact_n__less__m__mult__n, fact_ln__eq__zero__iff, fact_ln__inj__iff, fact_Suc__mono, fact_zero__less__power, arity_Nat__Onat__Rings_Ocomm__semiring__0, fact_psize__eq__0__iff, arity_Nat__Onat__Groups_Oordered__comm__monoid__add, fact_less__nat__zero__code, arity_RealDef__Oreal__Rings_Osemiring__0, arity_RealDef__Oreal__Rings_Olinordered__ring, fact_linorder__not__le, arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le, fact_mult__le__less__imp__less, fact_zadd__commute, fact_BseqD, fact_natceiling__le, fact_ln__less__cancel__iff, arity_Nat__Onat__Groups_Omonoid__mult, fact_diff__Suc__less, fact_complex__add, fact_power__increasing, fact_abs__triangle__ineq2__sym, fact_order__less__asym_H, arity_Polynomial__Opoly__Rings_Olinordered__ring__strict, fact_norm__add__less, arity_Nat__Onat__Rings_Ozero__neq__one, fact_div__nonpos__pos__le0, arity_Polynomial__Opoly__Groups_Oab__semigroup__mult, fact_linorder__antisym__conv2, fact_zero__less__diff, fact_mult__mono, fact_complex__Re__le__cmod, fact_leI, fact_field__power__not__zero, fact_natceiling__le__eq__one, arity_RealDef__Oreal__Rings_Oidom, fact_Suc__le__mono, fact_mult__poly__0__left, fact_real__root__pos__mult__exp, fact_zero__le__natfloor, fact_order__eq__refl, fact_zero__le__zpower__abs, fact_add__right__imp__eq, fact_realpow__two__diff, arity_Complex__Ocomplex__Groups_Oab__semigroup__mult, arity_Polynomial__Opoly__Rings_Oidom, fact_BseqI, fact_diff__diff__cancel, fact_mult_Ononneg__bounded, fact_cos__one__sin__zero, fact_real__root__le__mono, fact_nat__add__right__cancel, fact_gr0__conv__Suc, arity_Polynomial__Opoly__Groups_Ominus, fact_linorder__neqE, fact_o__eq__elim, arity_RealDef__Oreal__Groups_Omonoid__add, fact_div__le__dividend, fact_cos__n__Re__cis__pow__n, fact_abs__diff__triangle__ineq, fact_mult__le__mono2, fact_Suc__diff__diff, fact_mult_Oadd__right, fact_le__neq__implies__less, arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add, arity_Polynomial__Opoly__Rings_Ocomm__ring, fact_zle__linear, fact_lemma__NBseq__def2, fact_zero__le__one, fact_cis__zero, fact_nat__add__left__cancel__less, fact_int__0__less__1, fact_o__apply, fact_complex__Im__i, help_c__COMBC__1, fact_mult__left_Ononneg__bounded, fact_real__root__pos2, fact_real__root__mult__exp, fact_le__add__iff2, fact_mult__eq__self__implies__10, fact_complex_Osize_I2_J, fact_mult__left__le__imp__le, fact_add__pos__nonneg, fact_complex__Re__mult, fact_complex__Im__one, arity_Int__Oint__Orderings_Oord, arity_RealDef__Oreal__Groups_Ocancel__comm__monoid__add, fact_sin__cos__squared__add3, arity_RealDef__Oreal__Rings_Ocomm__ring__1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J, fact_linorder__antisym__conv3, fact_add__strict__increasing2, fact_order__less__imp__not__eq, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J, fact_linorder__neqE__linordered__idom, fact_lemma__realpow__diff, arity_Polynomial__Opoly__Rings_Olinordered__semidom, fact_mult__strict__left__mono, fact_add__le__imp__le__left, arity_RealDef__Oreal__Fields_Ofield__inverse__zero, fact_zero__le__mult__iff, arity_Int__Oint__Groups_Ozero, fact_add__nonneg__eq__0__iff, fact_real__of__nat__zero__iff, arity_Complex__Ocomplex__Rings_Ocomm__semiring__1, arity_Polynomial__Opoly__Orderings_Olinorder, fact_seq__suble, fact_div__1, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J, fact_div__if, fact_abs__triangle__ineq2, fact_Re_Opos__bounded, fact_zadd__left__mono, fact_sum__squares__le__zero__iff, arity_Int__Oint__Rings_Olinordered__semiring__1__strict, fact_mult__right_Ozero, fact_mult__le__cancel__left__neg, fact_real__root__pow__pos2, fact_sum__squares__ge__zero, fact_add__increasing2, fact_split__div_H, fact_natceiling__mono, fact_linorder__antisym__conv1, arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add, fact_zdiv__zero, fact_mult_Ozero__left, fact_real__mult__commute, fact_split__div__lemma, fact_real__root__strict__decreasing, fact_real__root__eq__iff, fact_gr0I, fact_i__def, arity_Int__Oint__Rings_Olinordered__ring, fact_add__0__iff, fact_le__funD, fact_add__right__mono, fact_le__fun__def, arity_Int__Oint__Groups_Ocancel__comm__monoid__add, fact_nat__less__real__le, fact_real__of__nat__less__iff, fact_one__less__power, fact_power__strict__increasing, fact_real__mult__assoc, fact_one__le__power, fact_even__less__0__iff, fact_self__quotient__aux1, fact_complex__i__not__zero, fact_power__commutes, fact_power__real__of__nat, fact_le__natfloor__eq__one, arity_Int__Oint__Groups_Ocomm__monoid__add, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J, arity_RealDef__Oreal__Groups_Omonoid__mult, fact_add__0__left, fact_add__mono, 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% 62.42/32.50  % No. of inferences in proof: 2522
% 62.42/32.50  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 16898 ms resp. 15904 ms w/o parsing
% 66.46/33.33  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 66.80/33.35  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------