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

%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : LCL796-1 : TPTP v9.3.1. Released v4.1.0.
% 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 : n006.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 07:52:57 AM UTC 2026

% Result   : Unsatisfiable 28.77s 12.49s
% Output   : Refutation 32.14s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    2
%            Number of leaves      : 1004
% Syntax   : Number of formulae    : 2009 ( 391 unt;   0 typ;   0 def)
%            Number of atoms       : 4863 (1244 equ;   0 cnn)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives : 33008 (2702   ~;2854   |;   0   &;27452   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (  10 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  119 ( 117 usr;  13 con; 0-6 aty)
%            Number of variables   : 6170 (   0   ^;6170   !;   0   ?;6170   :)

% Comments : 
%------------------------------------------------------------------------------
thf(c_Type_Otyping_decl,type,
    c_Type_Otyping: $i > $i > $i > $o ).

thf(c_Type_Oshift_decl,type,
    c_Type_Oshift: $i > $i > $i > $i > $i ).

thf(v_e_____decl,type,
    v_e____: $i ).

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

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

thf(v_T_H_H_____decl,type,
    v_T_H_H____: $i ).

thf(tc_Type_Otype_decl,type,
    tc_Type_Otype: $i ).

thf(c_Lambda_OdB_OVar_decl,type,
    c_Lambda_OdB_OVar: $i > $i ).

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

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

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

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

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

thf(c_Orderings_Oord__class_Omax_decl,type,
    c_Orderings_Oord__class_Omax: $i > $i > $i > $i ).

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

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

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

thf(c_Orderings_Oord__class_Omin_decl,type,
    c_Orderings_Oord__class_Omin: $i > $i > $i > $i ).

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

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

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

thf(c_Nat_Onat_Onat__case_decl,type,
    c_Nat_Onat_Onat__case: $i > $i > $i > $i > $i ).

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

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

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

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

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

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

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

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

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

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

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

thf(c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1_decl,type,
    c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1: $i > $i ).

thf(t_a_decl,type,
    t_a: $i ).

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

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

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

thf(c_Lambda_OdB_OApp_decl,type,
    c_Lambda_OdB_OApp: $i ).

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

thf(c_Type_Otype_OFun_decl,type,
    c_Type_Otype_OFun: $i > $i > $i ).

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

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

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

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

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

thf(c_Nat_Osemiring__1__class_Oof__nat_decl,type,
    c_Nat_Osemiring__1__class_Oof__nat: $i > $i > $i ).

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

thf(class_Orderings_Obot_decl,type,
    class_Orderings_Obot: $i > $o ).

thf(c_Orderings_Obot__class_Obot_decl,type,
    c_Orderings_Obot__class_Obot: $i > $i ).

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

thf(c_List_Ofoldl_decl,type,
    c_List_Ofoldl: $i > $i > $i > $i > $i > $i ).

thf(c_Lambda_OdB_OAbs_decl,type,
    c_Lambda_OdB_OAbs: $i > $i ).

thf(tc_Lambda_OdB_decl,type,
    tc_Lambda_OdB: $i ).

thf(c_Lambda_Obeta_decl,type,
    c_Lambda_Obeta: $i > $i > $o ).

thf(c_Lambda_Osubst_decl,type,
    c_Lambda_Osubst: $i > $i > $i > $i ).

thf(c_InductTermi_OIT_decl,type,
    c_InductTermi_OIT: $i > $o ).

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

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

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

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

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

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

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

thf(c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1_decl,type,
    c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1: $i > $i > $i ).

thf(c_Nat_Osemiring__1__class_Oof__nat__aux_decl,type,
    c_Nat_Osemiring__1__class_Oof__nat__aux: $i > $i > $i > $i > $i ).

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

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

thf(class_OrderedGroup_Osemigroup__add_decl,type,
    class_OrderedGroup_Osemigroup__add: $i > $o ).

thf(c_Lambda_OdB_OdB__case_decl,type,
    c_Lambda_OdB_OdB__case: $i > $i > $i > $i > $i > $i ).

thf(c_Lambda_OdB_OdB__size_decl,type,
    c_Lambda_OdB_OdB__size: $i > $i ).

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

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

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

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

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

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

thf(c_Lambda_Olift_decl,type,
    c_Lambda_Olift: $i > $i > $i ).

thf(class_Nat_Osemiring__char__0_decl,type,
    class_Nat_Osemiring__char__0: $i > $o ).

thf(c_List_Oreplicate_decl,type,
    c_List_Oreplicate: $i > $i > $i > $i ).

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

thf(c_Lambda_Oliftn_decl,type,
    c_Lambda_Oliftn: $i > $i > $i > $i ).

thf(c_Lambda_OdB_OdB__rec_decl,type,
    c_Lambda_OdB_OdB__rec: $i > $i > $i > $i > $i > $i ).

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

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

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

thf(c_Nat_Onat_Onat__rec_decl,type,
    c_Nat_Onat_Onat__rec: $i > $i > $i > $i > $i ).

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

thf(v_u_____decl,type,
    v_u____: $i ).

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

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

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

thf(c_List_Odrop_decl,type,
    c_List_Odrop: $i > $i > $i > $i ).

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

thf(c_Type_Osko__Type__Xtyping__elims__3__1_decl,type,
    c_Type_Osko__Type__Xtyping__elims__3__1: $i > $i > $i > $i ).

thf(c_Type_Osko__Type__Xtyping__elims__3__2_decl,type,
    c_Type_Osko__Type__Xtyping__elims__3__2: $i > $i > $i > $i ).

thf(c_Nitpick_Onat__gcd_decl,type,
    c_Nitpick_Onat__gcd: $i > $i > $i ).

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

thf(v_T_____decl,type,
    v_T____: $i ).

thf(v_sko__Nat__Xnat__Xinducts__1_decl,type,
    v_sko__Nat__Xnat__Xinducts__1: $i > $i ).

thf(v_sko__List__Xlexn__Xinduct__1_decl,type,
    v_sko__List__Xlexn__Xinduct__1: $i > $i ).

thf(v_sko__Nat__Xnat__Xinduct__1_decl,type,
    v_sko__Nat__Xnat__Xinduct__1: $i > $i ).

thf(c_ATP__Linkup_Osko__Nat__Xnat__induct__1__1_decl,type,
    c_ATP__Linkup_Osko__Nat__Xnat__induct__1__1: $i > $i ).

thf(c_ATP__Linkup_Osko__Nat__Xzero__induct__1__1_decl,type,
    c_ATP__Linkup_Osko__Nat__Xzero__induct__1__1: $i > $i ).

thf(v_sko__List__Xreplicate__Xinduct__1_decl,type,
    v_sko__List__Xreplicate__Xinduct__1: $i > $i ).

thf(v_sko__List__Xupt__Xinduct__1_decl,type,
    v_sko__List__Xupt__Xinduct__1: $i > $i ).

thf(c_nat_Onat__size_decl,type,
    c_nat_Onat__size: $i > $i ).

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

thf(c_Type_Osko__Type__Xabs__typeE__1__1_decl,type,
    c_Type_Osko__Type__Xabs__typeE__1__1: $i > $i > $i ).

thf(c_Type_Osko__Type__Xabs__typeE__1__2_decl,type,
    c_Type_Osko__Type__Xabs__typeE__1__2: $i > $i > $i ).

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

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

thf(tc_String_Ochar_decl,type,
    tc_String_Ochar: $i ).

thf(c_String_Ochar_Ochar__size_decl,type,
    c_String_Ochar_Ochar__size: $i > $i ).

thf(270,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__Suc__diff__eq2_0) ).

thf(1665,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(358,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__assoc_0) ).

thf(1879,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(167,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__mod_1) ).

thf(1411,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(191,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_2) ).

thf(1469,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(440,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit__asm_2) ).

thf(2072,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(522,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__less__iff__disj_0) ).

thf(2263,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(910,axiom,
    ! [A: $i] :
      ( ( tc_Lambda_OdB @ ( A @ c_Lambda_OdB_OVar @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osize_I4_J_0) ).

thf(3167,plain,
    ! [A: $i] :
      ( ( tc_Lambda_OdB @ ( A @ c_Lambda_OdB_OVar @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(605,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__least_0) ).

thf(2463,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(612,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__less__zero__iff__single__less__zero_0) ).

thf(2480,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(334,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__mult1__if_0) ).

thf(1818,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(390,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__gr__0_2) ).

thf(1956,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(754,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__leE_0) ).

thf(2822,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(673,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ c_Suc @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__antisym_0) ).

thf(2630,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ c_Suc @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(722,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__strict__left__mono__neg_0) ).

thf(2744,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(149,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div__lemma_1) ).

thf(1369,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(77,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I3_J_0) ).

thf(1190,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(536,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__gr__0_0) ).

thf(2295,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(995,axiom,
    tc_nat @ class_OrderedGroup_Oab__semigroup__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Oab__semigroup__mult) ).

thf(3345,plain,
    tc_nat @ class_OrderedGroup_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(555,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_n__less__n__mult__m_0) ).

thf(2341,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(553,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__add__left__eq_0) ).

thf(2336,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(831,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gr__implies__not0_0) ).

thf(3008,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(455,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( B @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ c_Suc @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I28_J_0) ).

thf(2109,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( B @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ c_Suc @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(184,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__double__add__iff__zero__le__single__add_0) ).

thf(1454,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(422,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__self2_0) ).

thf(2031,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(740,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__is__add_1) ).

thf(2787,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(658,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__by__0_0) ).

thf(2595,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(72,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zmod__simps_I4_J_0) ).

thf(1179,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(551,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__1_0) ).

thf(2331,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(530,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__maxI1_0) ).

thf(2280,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(44,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__Suc__shift_0) ).

thf(1110,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(816,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_0) ).

thf(2976,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(982,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__trans_0) ).

thf(3317,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(679,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OAbs ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I8_J_0) ).

thf(2644,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OAbs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(656,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__gt1__lemma_0) ).

thf(2590,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(693,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ c_Suc @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div__lemma_0) ).

thf(2676,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ c_Suc @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(880,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__comm__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__comm__semiring__strict) ).

thf(3108,plain,
    tc_nat @ class_Ring__and__Field_Oordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(397,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__mono_0) ).

thf(1971,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(282,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__le__mono_0) ).

thf(1692,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(15,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neqE_0) ).

thf(1041,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(69,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__pos__nonneg_0) ).

thf(1173,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(564,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__pred_0) ).

thf(2363,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(81,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__right__le__one__le_0) ).

thf(1200,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(443,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( F @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I20_J_0) ).

thf(2077,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( F @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(863,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__diff_0) ).

thf(3072,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(377,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__gr__0_1) ).

thf(1924,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(1,negated_conjecture,
    ~ ( v_T_H_H____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Lambda_OdB_OVar @ ( tc_Type_Otype @ ( v_T_H_H____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conjecture_0) ).

thf(1005,plain,
    ~ ( v_T_H_H____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Lambda_OdB_OVar @ ( tc_Type_Otype @ ( v_T_H_H____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).

thf(500,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__iff__diff__less__0_0) ).

thf(2210,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(118,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__eq__1__iff_0) ).

thf(1290,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(359,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__one_0) ).

thf(1882,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(848,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__self_0) ).

thf(3040,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(904,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Nat_Osemiring__1__class_Oof__nat__aux ) ) ) )
        = C )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__aux_Osimps_I1_J_0) ).

thf(3153,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Nat_Osemiring__1__class_Oof__nat__aux ) ) ) )
        = C )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(232,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Obot )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_fun__Orderings_Obot) ).

thf(1569,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Obot )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(383,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ c_Suc @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__less__Suc_0) ).

thf(1939,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ c_Suc @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(452,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__left_0) ).

thf(2099,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(531,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__lessD_0) ).

thf(2282,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(457,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__distrib_0) ).

thf(2114,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(56,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__eq__cancel1_0) ).

thf(1142,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(202,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( D @ c_Suc @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osplit__asm_1) ).

thf(1497,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( D @ c_Suc @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(887,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__not__refl_0) ).

thf(3120,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(697,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__ge__zero_0) ).

thf(2686,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(894,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__cancel1_0) ).

thf(3135,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(489,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__mono1_0) ).

thf(2184,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(708,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__self__is__m_0) ).

thf(2712,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(94,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__mono_0) ).

thf(1233,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(138,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__Suc2_0) ).

thf(1342,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(427,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__mult__distrib_0) ).

thf(2042,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(143,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_semiring__div__class_Omod__div__equality_H_0) ).

thf(1354,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(186,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__add__cong_0) ).

thf(1458,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(321,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I10_J_0) ).

thf(1788,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(933,axiom,
    tc_nat @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add) ).

thf(3217,plain,
    tc_nat @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(852,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinducts__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Oinducts_0) ).

thf(3049,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinducts__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(55,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( B @ ( A @ c_Type_Otype_OFun ) )
       != ( D @ ( C @ c_Type_Otype_OFun ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_type_Osimps_I2_J_1) ).

thf(1137,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( B @ ( A @ c_Type_Otype_OFun ) )
       != ( D @ ( C @ c_Type_Otype_OFun ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(65,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__cases_0) ).

thf(1164,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(408,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ c_Lambda_Obeta ) )
      | ( C @ ( B @ ( D @ c_Lambda_Osubst ) ) @ ( C @ ( B @ ( A @ c_Lambda_Osubst ) ) @ c_Lambda_Obeta ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__preserves__beta_0) ).

thf(1999,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ c_Lambda_Obeta ) )
      | ( C @ ( B @ ( D @ c_Lambda_Osubst ) ) @ ( C @ ( B @ ( A @ c_Lambda_Osubst ) ) @ c_Lambda_Obeta ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(599,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Opwr__pwr_0) ).

thf(2448,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(992,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__eq__cancel__disj_0) ).

thf(3338,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(610,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I3_J_0) ).

thf(2476,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(481,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__mul__solve_1) ).

thf(2168,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(375,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__eq_0) ).

thf(1919,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(952,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1 ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ex__least__nat__le_2) ).

thf(3253,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1 ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(603,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__iff2_1) ).

thf(2459,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(24,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I24_J_0) ).

thf(1063,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(389,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__less__iff__conj_1) ).

thf(1954,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(763,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__SucI_0) ).

thf(2844,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(79,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__0__less__mult__iff_0) ).

thf(1196,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(429,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__neg_0) ).

thf(2046,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(179,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nat_Odiff__diff__eq_0) ).

thf(1441,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(872,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0__less__iff_1) ).

thf(3088,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(264,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__le_0) ).

thf(1650,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(896,axiom,
    tc_nat @ class_Orderings_Obot,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Obot) ).

thf(3140,plain,
    tc_nat @ class_Orderings_Obot,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(582,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__by__0_0) ).

thf(2405,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(361,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I6_J_0) ).

thf(1887,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(808,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__leD_0) ).

thf(2957,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

thf(305,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__self1__is__m_0) ).

thf(1746,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(878,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( C @ c_Lambda_OdB_OAbs @ ( A @ c_Type_Otyping ) ) )
      | ( C @ ( A @ ( B @ c_Type_Osko__Type__Xtyping__elims__3__2 ) ) @ ( C @ ( tc_Type_Otype @ ( C @ ( A @ ( B @ c_Type_Osko__Type__Xtyping__elims__3__1 ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_typing__elims_I3_J_1) ).

thf(3103,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( C @ c_Lambda_OdB_OAbs @ ( A @ c_Type_Otyping ) ) )
      | ( C @ ( A @ ( B @ c_Type_Osko__Type__Xtyping__elims__3__2 ) ) @ ( C @ ( tc_Type_Otype @ ( C @ ( A @ ( B @ c_Type_Osko__Type__Xtyping__elims__3__1 ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(503,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( B = D )
      | ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ c_Suc @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__inject__base_0) ).

thf(2216,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( B = D )
      | ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ c_Suc @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(942,axiom,
    tc_nat @ class_OrderedGroup_Opordered__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__comm__monoid__add) ).

thf(3236,plain,
    tc_nat @ class_OrderedGroup_Opordered__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(558,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__iff__sup_0) ).

thf(2348,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(998,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__eq__self__zero_0) ).

thf(3350,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(713,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__less__Suc__eq_0) ).

thf(2723,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(639,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__zero__right_0) ).

thf(2548,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(319,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__neq__trans_0) ).

thf(1782,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(780,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_n__less__m__mult__n_0) ).

thf(2886,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(145,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__asym_0) ).

thf(1359,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(580,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__le__divisor_0) ).

thf(2400,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(438,axiom,
    ! [A: $i] :
      ( ( A @ c_Suc )
     != A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__n__not__n_0) ).

thf(2067,plain,
    ! [A: $i] :
      ( ( A @ c_Suc )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(395,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__nonneg__nonpos2_0) ).

thf(1967,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(753,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__apps__eq__Var__apps__conv_1) ).

thf(2819,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(327,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__increasing__iff_1) ).

thf(1803,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(467,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__le__self_0) ).

thf(2136,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(369,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__strict__mono_H_0) ).

thf(1904,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(211,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I23_J_0) ).

thf(1518,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(106,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I6_J_0) ).

thf(1265,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(92,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__inf__iff_1) ).

thf(1228,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(254,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__eq__refl_0) ).

thf(1623,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(195,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__idem_0) ).

thf(1482,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(102,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__less_1) ).

thf(1254,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(662,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__imp__eq_0) ).

thf(2603,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(842,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1__no__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_field__power__not__zero_0) ).

thf(3029,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(689,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__less__iff__conj_1) ).

thf(2667,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(525,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__a_0) ).

thf(2269,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(727,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__mult__distrib_0) ).

thf(2755,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(355,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I18_J_0) ).

thf(1871,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(784,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__commute_0) ).

thf(2897,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(463,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__if_0) ).

thf(2126,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(648,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__absorb1_0) ).

thf(2571,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(683,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I1_J_0) ).

thf(2654,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(256,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__is__add_2) ).

thf(1628,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(949,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( B @ ( E @ c_Lambda_Osubst ) ) @ ( A @ c_Type_Otyping ) ) )
      | ~ ( F @ ( E @ ( tc_Type_Otype @ ( C @ ( D @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__lemma_0) ).

thf(3248,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( B @ ( E @ c_Lambda_Osubst ) ) @ ( A @ c_Type_Otyping ) ) )
      | ~ ( F @ ( E @ ( tc_Type_Otype @ ( C @ ( D @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(166,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__iff_0) ).

thf(1409,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(117,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I25_J_0) ).

thf(1287,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(935,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
        = B )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__Var_1) ).

thf(3221,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
        = B )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(718,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__one_0) ).

thf(2735,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(751,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__commute_0) ).

thf(2814,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(426,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__div__geq_0) ).

thf(2039,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(962,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Type_Otyping ) ) )
      | ( B @ ( A @ c_Type_Osko__Type__Xabs__typeE__1__2 ) @ ( B @ ( tc_Type_Otype @ ( B @ ( A @ c_Type_Osko__Type__Xabs__typeE__1__1 ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_abs__typeE_0) ).

thf(3275,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Type_Otyping ) ) )
      | ( B @ ( A @ c_Type_Osko__Type__Xabs__typeE__1__2 ) @ ( B @ ( tc_Type_Otype @ ( B @ ( A @ c_Type_Osko__Type__Xabs__typeE__1__1 ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

thf(268,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__right__eq_0) ).

thf(1660,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(121,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__Suc_0) ).

thf(1298,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(554,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__le__add__iff2_0) ).

thf(2339,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(850,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nat_Oadd__0__right_0) ).

thf(3046,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(597,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__max__iff__disj_1) ).

thf(2444,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(351,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I4_J_0) ).

thf(1860,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(131,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__c_0) ).

thf(1323,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(901,axiom,
    tc_nat @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Opreorder) ).

thf(3148,plain,
    tc_nat @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(208,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_Suc @ ( B @ c_Nat_Osemiring__1__class_Oof__nat__aux ) ) ) )
        = ( A @ ( D @ ( B @ hAPP ) @ ( C @ ( B @ c_Nat_Osemiring__1__class_Oof__nat__aux ) ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__aux_Osimps_I2_J_0) ).

thf(1511,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ c_Suc @ ( B @ c_Nat_Osemiring__1__class_Oof__nat__aux ) ) ) )
        = ( A @ ( D @ ( B @ hAPP ) @ ( C @ ( B @ c_Nat_Osemiring__1__class_Oof__nat__aux ) ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(527,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__Suc__less__one_0) ).

thf(2274,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Oone__class_Oone @ ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(695,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oless__supI1_0) ).

thf(2682,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(889,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__le_1) ).

thf(3125,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(337,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__le__zero__iff__single__add__le__zero_1) ).

thf(1826,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(235,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Omul__c_0) ).

thf(1577,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(965,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( B @ c_Lambda_OdB_OVar ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I7_J_0) ).

thf(3280,plain,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( B @ c_Lambda_OdB_OVar ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(405,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__strict__left__mono_0) ).

thf(1992,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(999,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring__0) ).

thf(3353,plain,
    tc_nat @ class_Ring__and__Field_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(22,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__iff__Suc__add_1) ).

thf(1059,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(849,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Zero__neq__Suc_0) ).

thf(3043,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(223,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__cancel_0) ).

thf(1545,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(376,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__strict__mono_0) ).

thf(1922,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(153,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__add_0) ).

thf(1379,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(58,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__le__mult__iff_2) ).

thf(1147,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(99,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__gt__zero__iff_2) ).

thf(1247,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(287,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__diff__iff_0) ).

thf(1704,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(732,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__mod__geq_0) ).

thf(2767,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(716,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__le__iff_0) ).

thf(2730,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(261,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ c_Type_Otype_OFun ) )
       != ( D @ ( C @ c_Type_Otype_OFun ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_type_Osimps_I2_J_0) ).

thf(1641,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ c_Type_Otype_OFun ) )
       != ( D @ ( C @ c_Type_Otype_OFun ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(822,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div_5) ).

thf(2989,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(574,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__strict__increasing__iff_0) ).

thf(2388,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(40,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Suc @ ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__diff__diff_0) ).

thf(1100,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ c_Suc @ ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(959,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__neq__one_0) ).

thf(3268,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero )
       != ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(757,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__less__eq__dividend_0) ).

thf(2829,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(33,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_4) ).

thf(1087,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(209,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__0__less__mult__iff_1) ).

thf(1514,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(194,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__diff__right_0) ).

thf(1479,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(248,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__trans__Suc_0) ).

thf(1609,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(161,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__le__iff__disj_2) ).

thf(1398,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(870,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__is__0_2) ).

thf(3084,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(46,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__eq__add__iff2_0) ).

thf(1115,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(675,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__mono1_0) ).

thf(2635,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(979,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Oinduct_0) ).

thf(3312,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(380,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__add_0) ).

thf(1931,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(486,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oless__supI2_0) ).

thf(2178,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(198,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__max__iff__disj_0) ).

thf(1489,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(189,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__distrib2_0) ).

thf(1465,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(291,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( D @ ( F @ ( E @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( D @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I4_J_0) ).

thf(1714,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( D @ ( F @ ( E @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( D @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(925,axiom,
    tc_nat @ class_OrderedGroup_Osemigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Osemigroup__add) ).

thf(3199,plain,
    tc_nat @ class_OrderedGroup_Osemigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(865,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xnat__induct__1__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__induct_1) ).

thf(3075,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xnat__induct__1__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(448,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) )
       != ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) ) )
      | ~ ( A @ class_Nat_Osemiring__char__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__eq__iff_0) ).

thf(2090,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) )
       != ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) ) )
      | ~ ( A @ class_Nat_Osemiring__char__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(824,axiom,
    tc_nat @ class_Ring__and__Field_Opordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Opordered__semiring) ).

thf(2996,plain,
    tc_nat @ class_Ring__and__Field_Opordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(738,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__less__add1_0) ).

thf(2782,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(520,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__le__cancel1_1) ).

thf(2259,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(908,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__0R_0) ).

thf(3163,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(135,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__nonneg__nonneg_0) ).

thf(1334,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(778,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__eq__0__iff_0) ).

thf(2880,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(423,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( B @ ( B @ ( A @ c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1 ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ex__least__nat__le_0) ).

thf(2034,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( B @ ( B @ ( A @ c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1 ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(416,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__mod_5) ).

thf(2017,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(602,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__apps__neq__Abs__apps_0) ).

thf(2456,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(412,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__gt__zero__iff_0) ).

thf(2007,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(170,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__SucI_0) ).

thf(1418,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(600,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__mult__neg__le_1) ).

thf(2451,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(350,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__commute_0) ).

thf(1858,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(461,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__split__asm_1) ).

thf(2122,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(338,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__eq__0__iff_1) ).

thf(1828,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(400,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__add__self2_0) ).

thf(1978,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(7,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__diff__distrib__left_0) ).

thf(1020,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(1002,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Lambda_Oliftn ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn__0_0) ).

thf(3357,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Lambda_Oliftn ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(392,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ c_Suc @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__self4_0) ).

thf(1960,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ c_Suc @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(969,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xlexn__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lexn_Oinduct_0) ).

thf(3290,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xlexn__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(974,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__semiring) ).

thf(3300,plain,
    tc_nat @ class_Ring__and__Field_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(736,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__is__add_3) ).

thf(2777,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(903,axiom,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ocomm__semiring__1) ).

thf(3152,plain,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(435,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__eq__0__iff_1) ).

thf(2058,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(67,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__two_0) ).

thf(1169,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(86,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__le__trans_0) ).

thf(1215,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(589,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__trans_0) ).

thf(2424,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(159,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__inf__iff_2) ).

thf(1394,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(749,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__c_0) ).

thf(2809,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(176,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__le__add__iff1_1) ).

thf(1433,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(498,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__double__add__iff__zero__le__single__add_1) ).

thf(2205,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(399,axiom,
    ! [A: $i] :
      ( ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oone__class_Oone @ c_Power_Opower_Opower ) ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__power__power_0) ).

thf(1975,plain,
    ! [A: $i] :
      ( ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oone__class_Oone @ c_Power_Opower_Opower ) ) ) )
      | ~ ( A @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(730,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( A @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__imp__diff__is__add_1) ).

thf(2762,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( A @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(624,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__assoc_0) ).

thf(2508,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(811,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_4) ).

thf(2964,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(861,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__eq_0) ).

thf(3068,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(152,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__le__mono_0) ).

thf(1377,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(668,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_3) ).

thf(2618,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(759,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv2_0) ).

thf(2833,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(920,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__is__0__eq_0) ).

thf(3189,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(324,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit__asm_1) ).

thf(1796,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(846,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ c_InductTermi_OIT )
      | ~ ( G @ ( C @ ( tc_Type_Otype @ ( A @ ( E @ ( F @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( D @ c_InductTermi_OIT )
      | ~ ( A @ ( D @ ( F @ c_Type_Otyping ) ) )
      | ( E @ ( D @ ( C @ c_Lambda_Osubst ) ) @ c_InductTermi_OIT )
      | ( v_T____
       != ( B @ ( A @ c_Type_Otype_OFun ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_MI1_0) ).

thf(3036,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ c_InductTermi_OIT )
      | ~ ( G @ ( C @ ( tc_Type_Otype @ ( A @ ( E @ ( F @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( D @ c_InductTermi_OIT )
      | ~ ( A @ ( D @ ( F @ c_Type_Otyping ) ) )
      | ( E @ ( D @ ( C @ c_Lambda_Osubst ) ) @ c_InductTermi_OIT )
      | ( v_T____
       != ( B @ ( A @ c_Type_Otype_OFun ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(632,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_infinite__descent0_2) ).

thf(2530,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(214,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__diff__left_0) ).

thf(1526,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(468,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit_0) ).

thf(2137,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(513,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__1__right_0) ).

thf(2242,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(139,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ c_Suc @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I35_J_0) ).

thf(1345,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ c_Suc @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(725,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__left__commute_0) ).

thf(2750,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(164,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__strict__right__mono__neg_0) ).

thf(1405,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(42,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__inverse_0) ).

thf(1104,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(284,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__mult__le__cancel1_1) ).

thf(1697,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(931,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__iff__diff__eq__0_1) ).

thf(3211,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(576,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__less__mono_0) ).

thf(2392,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(475,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__antisym_0) ).

thf(2154,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(212,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__add__iff2_1) ).

thf(1521,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(918,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__rec ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__rec__0_0) ).

thf(3184,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__rec ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(819,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv2_1) ).

thf(2982,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(352,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__gt__zero__iff_1) ).

thf(1863,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( A @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(807,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__eq__zero__iff_1) ).

thf(2954,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(876,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) )
      | ( ( B @ ( A @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_typing__elims_I1_J_0) ).

thf(3097,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) )
      | ( ( B @ ( A @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(938,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__less_0) ).

thf(3229,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(833,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_minus__nat_Odiff__0_0) ).

thf(3011,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(609,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zmod__simps_I2_J_0) ).

thf(2473,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(479,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__less__iff__conj_2) ).

thf(2164,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(919,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__is__0__eq_1) ).

thf(3186,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(459,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__le__imp__le__exp_0) ).

thf(2118,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(449,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__iff1_0) ).

thf(2093,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(719,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__neg__neg_0) ).

thf(2737,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(436,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__le__zero__iff_0) ).

thf(2061,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(28,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I18_J_0) ).

thf(1074,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(246,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I9_J_0) ).

thf(1604,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(794,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__less__power_0) ).

thf(2922,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(975,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__is__0_1) ).

thf(3301,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(976,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__Suc0_0) ).

thf(3303,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(649,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_0) ).

thf(2574,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(178,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__div__trivial_0) ).

thf(1438,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(911,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xreplicate__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate_Oinduct_0) ).

thf(3169,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xreplicate__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(958,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__0_1) ).

thf(3265,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(541,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Onoteq__reduce_0) ).

thf(2307,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(855,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Nat_Osize__class_Osize ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__size_0) ).

thf(3054,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Nat_Osize__class_Osize ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(445,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel__left__neg_0) ).

thf(2083,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(300,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__1_5) ).

thf(1736,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(741,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__le__mono_0) ).

thf(2790,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(313,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__left_0) ).

thf(1768,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(354,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I13_J_0) ).

thf(1868,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(572,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I11_J_0) ).

thf(2382,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(17,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__self1_0) ).

thf(1047,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(298,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__cong_0) ).

thf(1730,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(21,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mod__equality_H_0) ).

thf(1057,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(111,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__mult__iff_3) ).

thf(1274,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(661,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__leE_0) ).

thf(2601,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(747,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supI_0) ).

thf(2805,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(516,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Divides_Omod__div__equality_H_0) ).

thf(2249,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(236,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__mono_0) ).

thf(1580,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(275,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__le__add2_0) ).

thf(1676,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(197,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__pos_0) ).

thf(1487,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(944,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_List_Oreplicate ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__replicate_0) ).

thf(3239,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_List_Oreplicate ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(663,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__less_1) ).

thf(2606,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(821,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc )
        = B )
      | ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__Suc_1) ).

thf(2986,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc )
        = B )
      | ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(631,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__infE_0) ).

thf(2528,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(929,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( B @ c_Lambda_OdB_OAbs ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I6_J_0) ).

thf(3207,plain,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( B @ c_Lambda_OdB_OAbs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(967,axiom,
    v_T_H_H____ @ ( v_a____ @ ( tc_Type_Otype @ ( v_T____ @ ( v_i____ @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_argT_0) ).

thf(3286,plain,
    v_T_H_H____ @ ( v_a____ @ ( tc_Type_Otype @ ( v_T____ @ ( v_i____ @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(367,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__less__imp__less__base_0) ).

thf(1900,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(295,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__add__iff1_0) ).

thf(1723,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(956,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gr0__conv__Suc_1) ).

thf(3261,plain,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(90,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I19_J_0) ).

thf(1223,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(87,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__le__eq_0) ).

thf(1217,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(472,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__decreasing_0) ).

thf(2147,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(839,axiom,
    ( ( tc_nat @ c_Orderings_Obot__class_Obot )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_bot__nat__def_0) ).

thf(3022,plain,
    ( ( tc_nat @ c_Orderings_Obot__class_Obot )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(204,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__less__cancel1_1) ).

thf(1502,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(336,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__split_1) ).

thf(1824,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(252,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult2__eq_0) ).

thf(1619,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(465,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__cancel_0) ).

thf(2132,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(585,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__eq__0__iff_0) ).

thf(2414,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(950,axiom,
    tc_nat @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Olinorder) ).

thf(3250,plain,
    tc_nat @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(96,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__diff__inverse_0) ).

thf(1238,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(415,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_3) ).

thf(2015,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(172,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__less__zero__iff__single__less__zero_1) ).

thf(1423,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(960,axiom,
    tc_nat @ class_Ring__and__Field_Ono__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ono__zero__divisors) ).

thf(3271,plain,
    tc_nat @ class_Ring__and__Field_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(728,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__Suc__less_0) ).

thf(2758,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(672,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__less_1) ).

thf(2628,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(593,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ c_Suc @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__Suc_0) ).

thf(2433,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( A @ ( D @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ c_Suc @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( D @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(873,axiom,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OVar @ c_Lambda_OdB_OdB__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osize_I1_J_0) ).

thf(3090,plain,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OVar @ c_Lambda_OdB_OdB__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(120,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__less__iff__conj_0) ).

thf(1296,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(280,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zmod__simps_I1_J_0) ).

thf(1687,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(63,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__commutes_0) ).

thf(1159,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(341,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__le_1) ).

thf(1836,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(165,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__mono2_0) ).

thf(1407,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(456,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__left__le__one__le_0) ).

thf(2112,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(4,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__diff__distrib__left_0) ).

thf(1012,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(11,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ c_Suc @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__Suc__diff__eq1_0) ).

thf(1032,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ c_Suc @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(245,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omax ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__minus__add__max_0) ).

thf(1602,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omax ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(687,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__less_0) ).

thf(2663,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(227,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I16_J_0) ).

thf(1555,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(756,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I1_J_0) ).

thf(2826,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(980,axiom,
    tc_nat @ class_Ring__and__Field_Omult__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Omult__zero) ).

thf(3314,plain,
    tc_nat @ class_Ring__and__Field_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(972,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__0_2) ).

thf(3297,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(860,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_lessequals ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__0__eq_0) ).

thf(3065,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_lessequals ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(775,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I22_J_0) ).

thf(2872,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(388,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__1_1) ).

thf(1951,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(30,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__imp__le_0) ).

thf(1079,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(549,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__right__mono_0) ).

thf(2327,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(51,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( D @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__eq__less_0) ).

thf(1128,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( D @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(617,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__1_4) ).

thf(2492,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(258,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__self1__is__0_0) ).

thf(1634,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(158,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ c_Lambda_Obeta ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ c_Lambda_Obeta ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__preserves__beta_0) ).

thf(1392,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ c_Lambda_Obeta ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ c_Lambda_Obeta ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(844,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
      = C ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_shift__eq_0) ).

thf(3033,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(302,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__less__trans_0) ).

thf(1740,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(372,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__left__cancel_0) ).

thf(1910,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(476,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__right__cancel_0) ).

thf(2157,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(310,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__gt1_0) ).

thf(1762,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(570,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ c_Suc @ ( B @ ( A @ c_Nat_Onat_Onat__rec ) ) ) )
      = ( D @ ( C @ ( B @ ( A @ c_Nat_Onat_Onat__rec ) ) ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__rec__Suc_0) ).

thf(2377,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ c_Suc @ ( B @ ( A @ c_Nat_Onat_Onat__rec ) ) ) )
      = ( D @ ( C @ ( B @ ( A @ c_Nat_Onat_Onat__rec ) ) ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(762,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I17_J_0) ).

thf(2841,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(707,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__eq__tail__conv_0) ).

thf(2709,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(239,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Osemigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__assoc_0) ).

thf(1587,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Osemigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(173,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__div__cancel1_0) ).

thf(1425,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(714,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__less__Suc__eq_1) ).

thf(2726,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(886,axiom,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add__imp__le) ).

thf(3119,plain,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(867,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__0L_0) ).

thf(3078,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(943,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_infinite__descent0_1) ).

thf(3237,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(47,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I2_J_0) ).

thf(1118,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(928,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power_Opower_Opower__0_0) ).

thf(3205,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(827,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Nitpick_Onat__gcd ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__gcd_Osimps_0) ).

thf(3001,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Nitpick_Onat__gcd ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(514,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__left__mono_0) ).

thf(2245,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(141,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__increasing2_0) ).

thf(1350,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(879,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__0_0) ).

thf(3105,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(241,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Osubst ) ) )
      = ( D @ ( C @ ( B @ c_Lambda_Osubst ) ) @ ( D @ ( C @ ( A @ c_Lambda_Osubst ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__App_0) ).

thf(1592,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Osubst ) ) )
      = ( D @ ( C @ ( B @ c_Lambda_Osubst ) ) @ ( D @ ( C @ ( A @ c_Lambda_Osubst ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(34,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__square_0) ).

thf(1089,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(221,axiom,
    ! [A: $i] :
      ( ( tc_Lambda_OdB @ ( A @ c_Lambda_OdB_OAbs @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_Lambda_OdB @ ( A @ c_Nat_Osize__class_Osize ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osize_I6_J_0) ).

thf(1541,plain,
    ! [A: $i] :
      ( ( tc_Lambda_OdB @ ( A @ c_Lambda_OdB_OAbs @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_Lambda_OdB @ ( A @ c_Nat_Osize__class_Osize ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(108,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__strict__mono_0) ).

thf(1270,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(347,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__diff__eq_0) ).

thf(1851,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(315,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__leE_1) ).

thf(1772,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(104,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A
        = ( B @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__SucE_0) ).

thf(1259,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A
        = ( B @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(888,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__lt_0) ).

thf(3122,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(218,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__Suc__less_0) ).

thf(1534,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(323,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonpos__neg_0) ).

thf(1794,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(6,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neqE__ordered__idom_0) ).

thf(1017,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(815,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__d_0) ).

thf(2973,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(48,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__if_1) ).

thf(1121,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(934,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__0__right_0) ).

thf(3218,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(349,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__one__less__zero_0) ).

thf(1856,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(38,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_1) ).

thf(1096,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(273,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__mult__iff_1) ).

thf(1672,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(293,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I7_J_0) ).

thf(1719,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(769,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__Suc__Suc_0) ).

thf(2858,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(635,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__mult__iff_0) ).

thf(2537,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(858,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__Nat__Xnat__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Oinduct_1) ).

thf(3060,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__Nat__Xnat__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(914,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring) ).

thf(3176,plain,
    tc_nat @ class_Ring__and__Field_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(837,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I10_J_0) ).

thf(3018,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(133,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__less__power__Suc_0) ).

thf(1329,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(419,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__sup__iff_0) ).

thf(2023,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(146,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I14_J_0) ).

thf(1361,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(187,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__right__mono__neg_0) ).

thf(1461,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(773,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oless__infI1_0) ).

thf(2868,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(587,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__less__iff__conj_0) ).

thf(2419,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(444,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__mult_0) ).

thf(2080,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(362,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__double__add__iff__zero__less__single__add_1) ).

thf(1889,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(200,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__imp__of__nat_0) ).

thf(1493,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(744,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel__left__pos_1) ).

thf(2798,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(278,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__mono_0) ).

thf(1683,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(897,axiom,
    tc_nat @ class_Ring__and__Field_Omult__mono1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Omult__mono1) ).

thf(3141,plain,
    tc_nat @ class_Ring__and__Field_Omult__mono1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(777,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__0_0) ).

thf(2877,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(394,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I12_J_0) ).

thf(1964,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(385,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__eq__zero__iff_0) ).

thf(1942,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(644,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I11_J_0) ).

thf(2561,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(930,axiom,
    v_T____ @ ( v_u____ @ ( v_e____ @ c_Type_Otyping ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_uT_0) ).

thf(3210,plain,
    v_T____ @ ( v_u____ @ ( v_e____ @ c_Type_Otyping ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(758,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__self1_0) ).

thf(2830,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(73,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__mono_H_0) ).

thf(1182,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(543,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Oadd__a_0) ).

thf(2312,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(832,axiom,
    ! [B: $i,A: $i] : ( B @ ( A @ hAPP ) @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_typing_OVar_0) ).

thf(3010,plain,
    ! [B: $i,A: $i] : ( B @ ( A @ hAPP ) @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(168,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__Suc_0) ).

thf(1414,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(433,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__is__add_4) ).

thf(2053,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(494,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neq__iff_1) ).

thf(2196,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(512,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__strict__increasing__iff_1) ).

thf(2240,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(382,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__mod_0) ).

thf(1937,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(666,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__mono_0) ).

thf(2613,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(843,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring__1) ).

thf(3032,plain,
    tc_nat @ class_Ring__and__Field_Osemiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(322,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
       != B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__iff__inf_1) ).

thf(1791,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
       != B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(483,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__iff2_0) ).

thf(2172,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(425,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__cube_0) ).

thf(2038,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(181,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__le__add__iff1_0) ).

thf(1446,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( D @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(640,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__commute_0) ).

thf(2551,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(83,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__square__less__zero_0) ).

thf(1207,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(446,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel_0) ).

thf(2085,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(817,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__irrefl_0) ).

thf(2978,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(681,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__add__distrib__left_0) ).

thf(2649,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(14,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__add__iff2_0) ).

thf(1039,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( D @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(971,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__cancel2_0) ).

thf(3294,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(304,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel1_1) ).

thf(1744,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(946,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__0R_0) ).

thf(3242,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(622,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__iff2_0) ).

thf(2504,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( E @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( E @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(798,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__add_0) ).

thf(2932,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(628,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__assoc_0) ).

thf(2519,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(52,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I2_J_0) ).

thf(1131,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(271,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ c_Suc @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__le__imp__le__base_0) ).

thf(1668,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ c_Suc @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(766,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( B @ ( A @ c_Lambda_Olift ) @ c_Lambda_Osubst ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__lift_0) ).

thf(2851,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( B @ ( A @ c_Lambda_Olift ) @ c_Lambda_Osubst ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(561,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__cancel__21_0) ).

thf(2355,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(619,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__inject__exp_0) ).

thf(2497,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(646,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__conv_0) ).

thf(2566,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(402,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__self2__is__id_0) ).

thf(1984,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(771,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OVar )
        = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__eq__apps__conv_0) ).

thf(2862,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OVar )
        = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(990,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__iff__diff__eq__0_0) ).

thf(3333,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(526,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__pos__pos_0) ).

thf(2272,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(567,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oless__infI2_0) ).

thf(2370,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(183,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I20_J_0) ).

thf(1451,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(540,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I4_J_0) ).

thf(2304,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(216,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__neg_1) ).

thf(1530,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(220,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__0__iff_2) ).

thf(1539,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(760,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__right__less__imp__less_0) ).

thf(2836,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(185,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__right__mono_0) ).

thf(1456,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(154,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I5_J_0) ).

thf(1382,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(615,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__0_0) ).

thf(2487,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(659,axiom,
    v_u____ @ c_InductTermi_OIT,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_uIT_0) ).

thf(2598,plain,
    v_u____ @ c_InductTermi_OIT,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(1000,axiom,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_neq0__conv_1) ).

thf(3354,plain,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(785,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_5) ).

thf(2900,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(339,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( D @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_left__add__mult__distrib_0) ).

thf(1831,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( D @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(23,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( F @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__pwr_0) ).

thf(1060,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( F @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(18,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__le__mono2_0) ).

thf(1050,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(652,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_5) ).

thf(2581,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(606,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__0_0) ).

thf(2465,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(356,axiom,
    ! [A: $i] :
      ( A
     != ( A @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_n__not__Suc__n_0) ).

thf(1874,plain,
    ! [A: $i] :
      ( A
     != ( A @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(884,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__1_0) ).

thf(3115,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(613,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__div__equality2_0) ).

thf(2482,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(244,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Suc @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__Suc_0) ).

thf(1599,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_Suc @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(702,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div__lemma_2) ).

thf(2697,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(881,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ( E @ ( B @ ( D @ c_Lambda_Olift ) @ ( tc_Type_Otype @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__type_0) ).

thf(3109,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ( E @ ( B @ ( D @ c_Lambda_Olift ) @ ( tc_Type_Otype @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(864,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__semiring__strict,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__semiring__strict) ).

thf(3074,plain,
    tc_nat @ class_Ring__and__Field_Oordered__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(829,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__semidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__semidom) ).

thf(3005,plain,
    tc_nat @ class_Ring__and__Field_Oordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(900,axiom,
    tc_nat @ class_Nat_Osemiring__char__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Nat_Osemiring__char__0) ).

thf(3147,plain,
    tc_nat @ class_Nat_Osemiring__char__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(129,axiom,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__add__less1_0) ).

thf(1318,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(332,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__0__iff_1) ).

thf(1814,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(318,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__eqI_1) ).

thf(1779,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(964,axiom,
    ! [A: $i] : ( A @ c_Lambda_OdB_OVar @ c_InductTermi_OIT ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__IT_0) ).

thf(3279,plain,
    ! [A: $i] : ( A @ c_Lambda_OdB_OVar @ c_InductTermi_OIT ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(779,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I23_J_0) ).

thf(2883,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(230,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__add__eq_0) ).

thf(1563,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(724,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__0__iff_0) ).

thf(2748,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(791,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__strict__right__mono_0) ).

thf(2915,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(505,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__right__mono_0) ).

thf(2221,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(251,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__sup__absorb_0) ).

thf(1616,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(407,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I13_J_0) ).

thf(1996,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(789,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__left__eq_0) ).

thf(2909,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(82,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Suc )
       != ( B @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__inject_0) ).

thf(1202,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Suc )
       != ( B @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(497,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__mult__distrib2_0) ).

thf(2203,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(97,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__le_2) ).

thf(1241,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(704,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( B @ c_Lambda_OdB_OVar @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_InductTermi_OIT ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_app__Var__IT_0) ).

thf(2702,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( B @ c_Lambda_OdB_OVar @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_InductTermi_OIT ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(345,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel__left__pos_0) ).

thf(1846,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(20,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__iff__gr__or__eq_0) ).

thf(1054,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(290,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__mult__less__cancel1_0) ).

thf(1712,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(977,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( E @ c_HOL_Oord__class_Oless ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
        = ( E @ ( A @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_shift__gt_0) ).

thf(3306,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( E @ c_HOL_Oord__class_Oless ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
        = ( E @ ( A @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(984,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__cancel2_1) ).

thf(3321,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(12,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ c_Suc @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = ( C @ ( B @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__case__Suc_0) ).

thf(1035,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ c_Suc @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = ( C @ ( B @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(566,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit__asm_0) ).

thf(2368,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(27,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__eq__0__iff_2) ).

thf(1071,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(581,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__imp__diff__is__add_0) ).

thf(2402,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(583,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__absorb2_0) ).

thf(2408,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(269,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__le__add1_0) ).

thf(1663,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(9,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__eqI_1) ).

thf(1026,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(365,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv_0) ).

thf(1895,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(66,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__iff__diff__le__0_0) ).

thf(1167,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(409,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__trans_0) ).

thf(2001,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(297,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__1_2) ).

thf(1727,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(485,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ c_Suc @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( D @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div_H_7) ).

thf(2176,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ c_Suc @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( D @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(787,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Omul__d_0) ).

thf(2904,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(642,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__iff__diff__less__0_1) ).

thf(2556,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(654,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__add__distrib__left_0) ).

thf(2585,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(110,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Osubst ) ) @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Obeta ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_beta_0) ).

thf(1273,plain,
    ! [B: $i,A: $i] : ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Osubst ) ) @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Obeta ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(801,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__mult__iff_4) ).

thf(2941,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(224,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__less_0) ).

thf(1548,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(957,axiom,
    ! [A: $i] :
      ( ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0_0) ).

thf(3262,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(882,axiom,
    tc_nat @ class_Ring__and__Field_Omult__mono,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Omult__mono) ).

thf(3111,plain,
    tc_nat @ class_Ring__and__Field_Omult__mono,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(257,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I2_J_0) ).

thf(1631,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(994,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0__less__iff_0) ).

thf(3343,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(510,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__1__right_0) ).

thf(2235,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(471,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__1__left_0) ).

thf(2144,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(113,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Osubst ) ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ c_InductTermi_OIT )
      | ~ ( B @ c_InductTermi_OIT )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ c_InductTermi_OIT ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Beta_0) ).

thf(1278,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Osubst ) ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ c_InductTermi_OIT )
      | ~ ( B @ c_InductTermi_OIT )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ c_InductTermi_OIT ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(370,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__lessD1_0) ).

thf(1906,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(54,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__less__add2_0) ).

thf(1135,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(263,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__iff1_1) ).

thf(1648,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(311,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__less__cancel1_0) ).

thf(1764,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(892,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__Nat__Xnat__Xinducts__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Oinducts_1) ).

thf(3130,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__Nat__Xnat__Xinducts__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(308,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__1_0) ).

thf(1754,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(535,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__one_0) ).

thf(2292,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(518,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I12_J_0) ).

thf(2253,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(473,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Oadd__c_0) ).

thf(2149,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(954,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__List__Xupt__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_upt_Oinduct_1) ).

thf(3257,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__List__Xupt__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(710,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit_2) ).

thf(2717,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(88,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__eq__0__iff_1) ).

thf(1219,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(421,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__mul__solve_0) ).

thf(2028,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( B = D )
      | ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(122,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__add__eq2_0) ).

thf(1301,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(783,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__diff__le_0) ).

thf(2894,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(151,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I15_J_0) ).

thf(1374,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(61,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__eq__add__iff2_1) ).

thf(1154,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(340,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Opwr__Suc_0) ).

thf(1833,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(862,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Orecs_I1_J_0) ).

thf(3070,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(374,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( E @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I1_J_0) ).

thf(1916,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( E @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(953,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__less_0) ).

thf(3255,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(630,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__left__idem_0) ).

thf(2525,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(491,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__eq__add__iff1_1) ).

thf(2188,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(616,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__add__self1_0) ).

thf(2489,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(823,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( B @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I1_J_0) ).

thf(2991,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( B @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(163,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_1) ).

thf(1403,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(752,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__1__mult_0) ).

thf(2817,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(49,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__eq__Suc__le_0) ).

thf(1124,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(466,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lem0_0) ).

thf(2134,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(809,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__less__imp__less__exp_0) ).

thf(2959,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(814,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__diff__cancel_0) ).

thf(2970,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(2,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__eqI_0) ).

thf(1007,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(983,axiom,
    ! [A: $i] :
      ( A
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__r0__iff_1) ).

thf(3319,plain,
    ! [A: $i] :
      ( A
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(940,axiom,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add) ).

thf(3234,plain,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(430,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__less__eq_0) ).

thf(2048,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(800,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__self1__is__id_0) ).

thf(2938,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = C )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(507,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Suc )
       != ( B @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Oinject_0) ).

thf(2225,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Suc )
       != ( B @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(360,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I25_J_0) ).

thf(1884,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(871,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__List__Xreplicate__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate_Oinduct_1) ).

thf(3086,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__List__Xreplicate__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(307,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__neg__pos_0) ).

thf(1752,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(265,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( E @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I11_J_0) ).

thf(1652,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( E @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(379,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__less_0) ).

thf(1929,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(966,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_right__minus__eq_1) ).

thf(3283,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(692,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__neg__nonpos_0) ).

thf(2674,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(626,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__0__left_0) ).

thf(2514,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(647,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ c_Suc @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__0__Suc_0) ).

thf(2568,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ c_Suc @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(156,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_Suc @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__eq__eq_1) ).

thf(1388,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_Suc @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(961,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_right__minus__eq_0) ).

thf(3272,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(91,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__le__cancel1_0) ).

thf(1226,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(701,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__Suc__eq__le_1) ).

thf(2695,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(103,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I9_J_0) ).

thf(1256,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(174,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__self_0) ).

thf(1428,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(493,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__eqI_0) ).

thf(2193,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(126,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__right__disj_2) ).

thf(1312,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(989,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osimps_I2_J_0) ).

thf(3330,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero )
     != ( A @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(43,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__Suc_0) ).

thf(1107,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(706,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( A @ ( B @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__add__iff2_1) ).

thf(2706,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( A @ ( B @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(366,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( C = D )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__scale__eq__noteq_0) ).

thf(1897,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( C = D )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(562,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__inf__distrib2_0) ).

thf(2358,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(288,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_no__zero__divirors__neq0_0) ).

thf(1707,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(853,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__Suc_0) ).

thf(3051,plain,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(782,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I2_J_1) ).

thf(2891,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(150,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__0__iff_2) ).

thf(1371,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(866,axiom,
    tc_nat @ class_Ring__and__Field_Opordered__cancel__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Opordered__cancel__semiring) ).

thf(3077,plain,
    tc_nat @ class_Ring__and__Field_Opordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(799,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Omul__1_0) ).

thf(2935,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(299,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__Suc2_0) ).

thf(1733,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(596,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__le__not__le_2) ).

thf(2442,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(726,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( B @ ( A @ c_Lambda_Olift ) @ c_InductTermi_OIT ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__IT_0) ).

thf(2753,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( B @ ( A @ c_Lambda_Olift ) @ c_InductTermi_OIT ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(434,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__eq__iff_2) ).

thf(2055,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(36,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( D @ c_Suc @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osplit_1) ).

thf(1092,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( D @ c_Suc @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(480,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__pos_0) ).

thf(2166,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(35,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_infinite__descent0_0) ).

thf(1090,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(557,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel1_1) ).

thf(2346,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(563,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__right_1) ).

thf(2361,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(694,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = F )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__add__iff1_0) ).

thf(2679,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = F )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(678,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__mono_0) ).

thf(2642,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(502,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__right_0) ).

thf(2214,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(676,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__App__neq__Var__apps_0) ).

thf(2637,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(451,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__mono1_0) ).

thf(2097,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(532,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__1_0) ).

thf(2284,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(538,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I7_J_0) ).

thf(2299,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(690,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__zero__left_0) ).

thf(2669,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Omult__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(458,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__nonneg__nonpos_0) ).

thf(2116,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(895,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__le__linear_0) ).

thf(3138,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(281,axiom,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs @ c_Lambda_OdB_OdB__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_Lambda_OdB_OdB__size @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osize_I3_J_0) ).

thf(1690,plain,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs @ c_Lambda_OdB_OdB__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_Lambda_OdB_OdB__size @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(233,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mod__trivial_0) ).

thf(1571,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(70,axiom,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__add__less2_0) ).

thf(1175,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(84,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__mult_0) ).

thf(1209,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(712,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__square_0) ).

thf(2721,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(115,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__le__mono_1) ).

thf(1282,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(611,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__pos_1) ).

thf(2478,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(951,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_plus__nat_Oadd__0_0) ).

thf(3251,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(234,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I3_J_0) ).

thf(1574,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ c_HOL_Oone__class_Oone @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(329,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__add__self2_0) ).

thf(1806,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(735,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I35_J_0) ).

thf(2774,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(968,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__if_0) ).

thf(3287,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(802,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__conv_1) ).

thf(2943,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(803,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_Lambda_OdB_OAbs ) )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( C @ c_Lambda_OdB_OAbs ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__eq__Abs__conv_0) ).

thf(2945,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_Lambda_OdB_OAbs ) )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( C @ c_Lambda_OdB_OAbs ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(924,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osplit_0) ).

thf(3197,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(326,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0__le__iff_0) ).

thf(1801,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(396,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__mult__pos__le_1) ).

thf(1969,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(907,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Olift ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift_Osimps_I1_J_0) ).

thf(3160,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Olift ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(124,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_even__less__0__iff_0) ).

thf(1307,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(487,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__0__iff_0) ).

thf(2180,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(755,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I5_J_0) ).

thf(2824,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(357,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__Suc__eq__Suc__mod_0) ).

thf(1877,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(721,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__le__zero__iff__single__add__le__zero_0) ).

thf(2742,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(137,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__0_0) ).

thf(1339,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(64,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel1_2) ).

thf(1162,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(598,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel1_2) ).

thf(2446,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(909,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__cancel1_1) ).

thf(3165,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(921,axiom,
    tc_nat @ class_Ring__and__Field_Ozero__neq__one,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ozero__neq__one) ).

thf(3192,plain,
    tc_nat @ class_Ring__and__Field_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(342,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__le_2) ).

thf(1838,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(59,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__self2_0) ).

thf(1149,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(71,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__le__mono_0) ).

thf(1177,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(604,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__div__cancel__disj_0) ).

thf(2461,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(550,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( C @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I12_J_0) ).

thf(2329,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( C @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(45,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__add__eq1_0) ).

thf(1112,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(420,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__iff__inf_0) ).

thf(2025,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(750,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__is__add_5) ).

thf(2812,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(464,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__mult1_0) ).

thf(2129,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(556,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__mult2_0) ).

thf(2343,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(671,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__right__cancel_0) ).

thf(2625,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(737,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__pos__neg_0) ).

thf(2780,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(524,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__mult__pos__le_0) ).

thf(2267,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(521,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__le__mult__iff_0) ).

thf(2261,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(335,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_1) ).

thf(1821,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(10,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__eqI_1) ).

thf(1029,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(830,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__self__eq__0_0) ).

thf(3006,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(637,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I8_J_0) ).

thf(2542,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(680,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__assoc_0) ).

thf(2647,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(119,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__le__zero__iff_1) ).

thf(1293,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(529,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__one__le__zero_0) ).

thf(2278,plain,
    ! [A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(68,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I4_J_0) ).

thf(1171,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(378,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I11_J_0) ).

thf(1926,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(501,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__le_0) ).

thf(2212,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(469,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I5_J_0) ).

thf(2139,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(674,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv2_0) ).

thf(2633,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(499,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__absorb0_0) ).

thf(2207,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(623,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__diff_0) ).

thf(2506,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(733,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( A @ c_Suc @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__Suc__eq_2) ).

thf(2770,plain,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( A @ c_Suc @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(442,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__cancel2_0) ).

thf(2075,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(80,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__mult__pos_0) ).

thf(1198,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(343,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_monoid__add__class_Oadd__0__right_0) ).

thf(1841,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(160,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv2_1) ).

thf(1396,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(595,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mod__equality2_0) ).

thf(2439,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(242,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( B @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__mul_0) ).

thf(1594,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( B @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(285,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__sup__distrib2_0) ).

thf(1699,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(432,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__less__iff__disj_1) ).

thf(2051,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(731,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__asym_H_0) ).

thf(2765,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(16,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I17_J_0) ).

thf(1044,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(25,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__cancel_0) ).

thf(1066,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(215,axiom,
    ! [A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( A @ c_Lambda_OdB_OAbs @ c_InductTermi_OIT ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Lambda_0) ).

thf(1528,plain,
    ! [A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( A @ c_Lambda_OdB_OAbs @ c_InductTermi_OIT ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(669,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( C @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__left__commute_0) ).

thf(2620,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( C @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(906,axiom,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocomm__monoid__add) ).

thf(3159,plain,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(437,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__less__linear_0) ).

thf(2064,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(5,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Oorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_fun__Orderings_Oorder) ).

thf(1015,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Oorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(945,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__imp__le_0) ).

thf(3241,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(836,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__1_0) ).

thf(3016,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(657,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__cancel_0) ).

thf(2592,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(177,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__0__iff_1) ).

thf(1435,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(987,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A = C )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__Var_2) ).

thf(3325,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A = C )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubst ) ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(898,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc )
       != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__Suc_0) ).

thf(3142,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Suc @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_Suc )
       != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(634,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_pos__add__strict_0) ).

thf(2535,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(453,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__eq__replicate_0) ).

thf(2101,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(147,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__left__commute_0) ).

thf(1364,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(813,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__diff__right__eq_0) ).

thf(2967,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(575,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__self3_0) ).

thf(2390,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_nat @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(100,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__greatest_0) ).

thf(1250,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(916,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__diff_0) ).

thf(3178,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(41,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I8_J_0) ).

thf(1102,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(283,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__eq__add__iff1_0) ).

thf(1694,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(700,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__infI1_0) ).

thf(2693,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(820,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__div__cancel_0) ).

thf(2984,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(398,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ c_Lambda_Obeta ) )
      | ( B @ ( C @ c_Lambda_Olift ) @ ( B @ ( A @ c_Lambda_Olift ) @ c_Lambda_Obeta ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__preserves__beta_0) ).

thf(1973,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ c_Lambda_Obeta ) )
      | ( B @ ( C @ c_Lambda_Olift ) @ ( B @ ( A @ c_Lambda_Olift ) @ c_Lambda_Obeta ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(328,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add1_0) ).

thf(1805,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(411,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__mult__neg__le_0) ).

thf(2005,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(387,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__mult__cancel1_0) ).

thf(1948,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(688,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_Suc @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__n__not__le__n_0) ).

thf(2665,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_Suc @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(190,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__infI_0) ).

thf(1467,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(682,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel2_1) ).

thf(2652,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(171,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__div__equality_0) ).

thf(1420,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(462,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_Suc @ c_Lambda_Oliftn ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_Lambda_Oliftn ) ) @ c_Lambda_Olift ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn__lift_0) ).

thf(2124,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_Suc @ c_Lambda_Oliftn ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_Lambda_Oliftn ) ) @ c_Lambda_Olift ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(391,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__mult__iff_5) ).

thf(1958,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(970,axiom,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Nat_Osize__class_Osize ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osize_I3_J_0) ).

thf(3292,plain,
    ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Nat_Osize__class_Osize ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(225,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__strict__decreasing_0) ).

thf(1551,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(330,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__add__iff2_0) ).

thf(1809,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(806,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__increasing_0) ).

thf(2952,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(123,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
       != C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__iff__sup_1) ).

thf(1304,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
       != C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(922,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__r0__iff_0) ).

thf(3193,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(545,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel2_2) ).

thf(2318,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(739,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__assoc_0) ).

thf(2784,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(31,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mod__eq__iff_1) ).

thf(1081,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_nat @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(796,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I2_J_0) ).

thf(2927,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(404,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__le__mono2_0) ).

thf(1990,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(851,axiom,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__Suc0_1) ).

thf(3048,plain,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(696,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__diff_0) ).

thf(2684,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(57,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__le1_0) ).

thf(1145,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(764,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__0__iff_3) ).

thf(2846,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(519,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__if_1) ).

thf(2256,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Odiv ) ) @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(645,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supI2_0) ).

thf(2564,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(325,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_combine__common__factor_0) ).

thf(1798,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(314,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__max__iff__disj_2) ).

thf(1770,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(709,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__left__le__imp__le_0) ).

thf(2715,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(578,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__left_1) ).

thf(2397,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(883,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( A @ ( E @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__0_0) ).

thf(3112,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( A @ ( E @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(686,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__le_1) ).

thf(2661,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(249,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I21_J_0) ).

thf(1611,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(162,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__one_0) ).

thf(1400,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(869,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__case__0_0) ).

thf(3082,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(932,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( B @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn_Osimps_I1_J_0) ).

thf(3214,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( B @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(655,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__max__iff__disj_1) ).

thf(2588,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(891,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__refl_0) ).

thf(3129,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(857,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__List__Xlexn__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lexn_Oinduct_1) ).

thf(3058,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ v_sko__List__Xlexn__Xinduct__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(210,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__Suc_0) ).

thf(1516,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(981,axiom,
    ! [A: $i] :
      ( ( tc_String_Ochar @ ( A @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_size__char_0) ).

thf(3315,plain,
    ! [A: $i] :
      ( ( tc_String_Ochar @ ( A @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(231,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div_1) ).

thf(1566,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( A @ ( D @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ hAPP ) @ hBOOL )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(414,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I7_J_0) ).

thf(2012,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(717,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I24_J_0) ).

thf(2732,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(101,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__right__le__imp__le_0) ).

thf(1252,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(93,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__1_0) ).

thf(1230,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(477,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( D @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Orecs_I3_J_0) ).

thf(2160,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( D @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(39,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__right_0) ).

thf(1098,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(781,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I6_J_0) ).

thf(2888,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(948,axiom,
    ! [A: $i] :
      ( ( A @ c_Suc @ c_nat_Onat__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_nat_Onat__size @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osize_I2_J_0) ).

thf(3246,plain,
    ! [A: $i] :
      ( ( A @ c_Suc @ c_nat_Onat__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( A @ c_nat_Onat__size @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(403,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__by__1_0) ).

thf(1987,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = B )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(927,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__is__0_0) ).

thf(3202,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(320,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__self2__is__0_0) ).

thf(1785,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(890,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ c_ATP__Linkup_Osko__Nat__Xnat__induct__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__induct_0) ).

thf(3127,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ c_ATP__Linkup_Osko__Nat__Xnat__induct__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(547,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__add__iff1_1) ).

thf(2322,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(136,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Opwr__mul_0) ).

thf(1336,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(607,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__le__mult__iff_1) ).

thf(2468,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(684,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__less__mono_0) ).

thf(2656,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(868,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xzero__induct__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__induct_1) ).

thf(3080,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xzero__induct__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(203,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__0_0) ).

thf(1499,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(259,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__le__iff__disj_0) ).

thf(1637,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(698,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__0__iff_4) ).

thf(2688,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(76,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__sup__iff_1) ).

thf(1188,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(75,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__le__power_0) ).

thf(1186,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(841,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neqE__nat_0) ).

thf(3026,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(591,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( A @ c_Lambda_Oliftn ) ) )
      = ( D @ ( C @ ( A @ c_Lambda_Oliftn ) ) @ ( D @ ( B @ ( A @ c_Lambda_Oliftn ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn_Osimps_I2_J_0) ).

thf(2429,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( A @ c_Lambda_Oliftn ) ) )
      = ( D @ ( C @ ( A @ c_Lambda_Oliftn ) ) @ ( D @ ( B @ ( A @ c_Lambda_Oliftn ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(105,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__inf__absorb_0) ).

thf(1262,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(95,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I15_J_0) ).

thf(1235,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(528,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__nonpos__nonpos_0) ).

thf(2276,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(504,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit_1) ).

thf(2219,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(638,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ab__semigroup__mult__class_Omult__ac_I1_J_0) ).

thf(2545,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(660,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__left__mono_0) ).

thf(2599,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(1003,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__neq__iff_0) ).

thf(3359,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(913,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_nat_Onat__size )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osize_I1_J_0) ).

thf(3174,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_nat_Onat__size )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(428,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__le2_0) ).

thf(2044,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(301,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_Lambda_OdB @ ( B @ c_Nat_Osize__class_Osize ) @ ( tc_Lambda_OdB @ ( A @ c_Nat_Osize__class_Osize ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osize_I5_J_0) ).

thf(1738,plain,
    ! [B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_Lambda_OdB @ ( B @ c_Nat_Osize__class_Osize ) @ ( tc_Lambda_OdB @ ( A @ c_Nat_Osize__class_Osize ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(875,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ~ ( B @ ( D @ c_Lambda_Obeta ) )
      | ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subject__reduction_0) ).

thf(3095,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ~ ( B @ ( D @ c_Lambda_Obeta ) )
      | ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(515,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__Suc__right_0) ).

thf(2247,plain,
    ! [B: $i,A: $i] :
      ( ( B @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(8,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__eqI_0) ).

thf(1023,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(1001,axiom,
    tc_nat @ class_OrderedGroup_Omonoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Omonoid__mult) ).

thf(3356,plain,
    tc_nat @ class_OrderedGroup_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(651,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__iff_0) ).

thf(2579,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(205,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( A @ c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1 ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ex__least__nat__le_1) ).

thf(1504,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( A @ c_ATP__Linkup_Osko__Nat__Xex__least__nat__le__1__1 ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(765,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( B @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__pwr_0) ).

thf(2848,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( B @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(381,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I10_J_0) ).

thf(1934,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(804,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Omul__a_0) ).

thf(2948,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(584,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__assoc2_0) ).

thf(2411,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(620,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__add1__eq_0) ).

thf(2500,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(222,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__imp__less__left_0) ).

thf(1543,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(312,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__lemma_0) ).

thf(1766,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(761,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__by__1_0) ).

thf(2838,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(643,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__div__decomp_0) ).

thf(2558,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(590,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( B @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I8_J_0) ).

thf(2426,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( B @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(219,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__add__right__eq_0) ).

thf(1536,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(401,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv3_0) ).

thf(1981,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(286,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__Suc__right_0) ).

thf(1702,plain,
    ! [B: $i,A: $i] :
      ( ( B @ c_Suc @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(571,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__0_0) ).

thf(2379,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(89,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Orderings_Obot__class_Obot @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_bot__least_0) ).

thf(1221,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Orderings_Obot__class_Obot @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(116,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I27_J_0) ).

thf(1284,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(196,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__eq__1__iff_2) ).

thf(1485,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(546,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__commute_0) ).

thf(2320,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(19,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__inf__iff_0) ).

thf(1052,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(206,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__less__0__iff_0) ).

thf(1506,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(306,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B = D )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__imp__eq__base_0) ).

thf(1749,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B = D )
      | ( ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(936,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_Nitpick_Onat__gcd ) )
        = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_Nitpick_Onat__gcd ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__gcd_Osimps_1) ).

thf(3224,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_Nitpick_Onat__gcd ) )
        = ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ ( B @ c_Nitpick_Onat__gcd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(238,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel2_0) ).

thf(1585,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(37,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__mult__pos2_0) ).

thf(1094,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(625,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__diff__left__eq_0) ).

thf(2511,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( A @ ( C @ ( A @ ( D @ ( B @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(272,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__mult__less__cancel1_1) ).

thf(1670,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(840,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Osubst ) ) @ c_InductTermi_OIT ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__Var__IT_0) ).

thf(3024,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ c_InductTermi_OIT )
      | ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Osubst ) ) @ c_InductTermi_OIT ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(885,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__diff_1) ).

thf(3117,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(770,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__Suc__eq__le_0) ).

thf(2860,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(715,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__0__less__mult__iff_2) ).

thf(2728,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(292,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__Suc__eq__0__disj_3) ).

thf(1717,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(748,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Olift ) )
      = ( C @ ( B @ c_Lambda_Olift ) @ ( C @ ( A @ c_Lambda_Olift ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift_Osimps_I2_J_0) ).

thf(2807,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Olift ) )
      = ( C @ ( B @ c_Lambda_Olift ) @ ( C @ ( A @ c_Lambda_Olift ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(109,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__Suc2_0) ).

thf(1272,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(905,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__antisym_0) ).

thf(3156,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(729,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I9_J_0) ).

thf(2760,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(795,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ab__semigroup__add__class_Oadd__ac_I1_J_0) ).

thf(2924,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(447,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__increasing2_0) ).

thf(2088,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(874,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ~ ( B @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_var__app__type__eq_0) ).

thf(3092,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ~ ( B @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(279,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omin ) ) @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__Suc__Suc_0) ).

thf(1685,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omin ) ) @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(226,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__le__iff__disj_1) ).

thf(1553,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(774,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__le__imp__less_0) ).

thf(2870,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(912,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__neq__zero_0) ).

thf(3171,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Oone__class_Oone )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(266,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I19_J_0) ).

thf(1654,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(50,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__mono2_0) ).

thf(1126,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(856,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) )
      = ( B @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__eq__replicate_2) ).

thf(3056,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) )
      = ( B @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(847,axiom,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ocomm__semiring) ).

thf(3039,plain,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(772,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__mult_0) ).

thf(2865,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(776,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__less__iff__conj_2) ).

thf(2875,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(384,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Odistrib__inf__le_0) ).

thf(1940,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(128,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__mult__distrib2_0) ).

thf(1316,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(32,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_no__zero__divisors_0) ).

thf(1084,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(62,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv1_1) ).

thf(1157,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(373,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__eq__less__or__eq_0) ).

thf(1913,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(542,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__mono1_0) ).

thf(2310,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(344,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__le_1) ).

thf(1844,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(665,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__assoc_0) ).

thf(2611,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(548,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__commute_0) ).

thf(2325,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(825,axiom,
    tc_nat @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add) ).

thf(2997,plain,
    tc_nat @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(182,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__idem_0) ).

thf(1448,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(996,axiom,
    tc_nat @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Divides_Osemiring__div) ).

thf(3346,plain,
    tc_nat @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(3,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Opreorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_fun__Orderings_Opreorder) ).

thf(1010,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Opreorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(157,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I10_J_0) ).

thf(1390,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(793,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Opwr__0_0) ).

thf(2919,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(353,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel1_0) ).

thf(1866,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(685,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn_Osimps_I1_J_1) ).

thf(2658,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(742,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I16_J_0) ).

thf(2792,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(788,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Odrop ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_List_Odrop ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__drop_0) ).

thf(2907,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Odrop ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_List_Odrop ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(262,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult1__eq_0) ).

thf(1646,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(132,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__assoc_0) ).

thf(1326,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(641,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__left_1) ).

thf(2554,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(991,axiom,
    ! [A: $i] :
      ( ( A @ c_String_Ochar_Ochar__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_char__size_0) ).

thf(3336,plain,
    ! [A: $i] :
      ( ( A @ c_String_Ochar_Ochar__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(348,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__mult__distrib_0) ).

thf(1854,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(213,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I9_J_0) ).

thf(1523,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(577,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__apps__eq__Var__apps__conv_0) ).

thf(2394,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(112,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__left__mono__neg_0) ).

thf(1276,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(240,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__mono_0) ).

thf(1590,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(155,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__neq__implies__less_0) ).

thf(1385,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(294,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__power_0) ).

thf(1721,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(484,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_even__less__0__iff_1) ).

thf(2174,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(193,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__a_0) ).

thf(1476,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(274,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel__left__disj_2) ).

thf(1674,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(768,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__add__self1_0) ).

thf(2855,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ c_HOL_Oone__class_Oone @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(260,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__maxI2_0) ).

thf(1639,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(767,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__mono1_0) ).

thf(2853,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Omult__mono1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(534,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__infI2_0) ).

thf(2290,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(517,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ c_Type_Otype_OFun ) @ ( B @ ( A @ c_Type_Otyping ) ) )
      | ~ ( E @ ( C @ ( A @ c_Type_Otyping ) ) )
      | ( D @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( A @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_App_0) ).

thf(2251,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ c_Type_Otype_OFun ) @ ( B @ ( A @ c_Type_Otyping ) ) )
      | ~ ( E @ ( C @ ( A @ c_Type_Otyping ) ) )
      | ( D @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( A @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(633,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Oadd__0_0) ).

thf(2532,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(207,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I22_J_0) ).

thf(1508,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(664,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__semiring__class_Odistrib_0) ).

thf(2608,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(941,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le0_0) ).

thf(3235,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(592,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__iff__gr__or__eq_1) ).

thf(2431,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(828,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Odrop ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__0_0) ).

thf(3003,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Odrop ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(98,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__apps__eq__Abs__apps__conv_0) ).

thf(1244,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(228,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__inf__distrib1_0) ).

thf(1558,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(474,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( F @ ( E @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( F @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( E @ ( D @ ( B @ hAPP ) @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Orecs_I2_J_0) ).

thf(2152,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( F @ ( E @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( F @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( E @ ( D @ ( B @ hAPP ) @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(413,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__mult__div__cancel__disj_1) ).

thf(2009,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_nat @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(199,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__iff1_1) ).

thf(1491,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ ( E @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(917,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__0__left_1) ).

thf(3181,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(986,axiom,
    tc_nat @ class_OrderedGroup_Oab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Oab__semigroup__add) ).

thf(3324,plain,
    tc_nat @ class_OrderedGroup_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(317,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__add__iff1_1) ).

thf(1777,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( D @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( C @ ( B @ ( tc_nat @ ( D @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(902,axiom,
    ! [A: $i] :
      ( ( A @ c_Suc )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osimps_I3_J_0) ).

thf(3149,plain,
    ! [A: $i] :
      ( ( A @ c_Suc )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(495,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Omul__pwr_0) ).

thf(2198,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(650,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ c_Suc )
        = B )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__lessI_0) ).

thf(2576,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ c_Suc )
        = B )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(78,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__idem_0) ).

thf(1193,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(247,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__le__zero__iff_2) ).

thf(1607,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(450,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__iff__diff__le__0_1) ).

thf(2095,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(544,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv1_0) ).

thf(2315,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(406,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__cancel1_0) ).

thf(1994,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(417,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Suc @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ ( A @ c_Nat_Osize__class_Osize ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osize_I4_J_0) ).

thf(2019,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Suc @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( tc_nat @ ( A @ c_Nat_Osize__class_Osize ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(790,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Opower__commutes_0) ).

thf(2912,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ ( A @ c_HOL_Oplus__class_Oplus @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(579,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__iff__add_1) ).

thf(2399,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(29,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__neg__neg_0) ).

thf(1077,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(478,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__iff_1) ).

thf(2162,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(488,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__le__dividend_0) ).

thf(2183,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(85,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__diff__inverse2_0) ).

thf(1212,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(786,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__less__iff_1) ).

thf(2902,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(460,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Onoteq__reduce_1) ).

thf(2120,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(508,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__cancel__21_1) ).

thf(2230,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(364,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oplus__class_Oplus @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Ogb__semiring__axioms_H_0) ).

thf(1893,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ ( A @ c_HOL_Otimes__class_Otimes @ ( A @ c_HOL_Oplus__class_Oplus @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(424,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__leI_0) ).

thf(2036,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(125,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_0) ).

thf(1309,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(720,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__neq__le__trans_0) ).

thf(2739,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(618,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I28_J_0) ).

thf(2494,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ c_Suc @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(180,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__eq__Suc__le_1) ).

thf(1444,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(792,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__nonneg_0) ).

thf(2917,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(670,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_Orderings_Oord__class_Omax ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omax ) ) @ c_Suc ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__Suc__Suc_0) ).

thf(2623,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ c_Suc @ ( A @ c_Suc @ c_Orderings_Oord__class_Omax ) ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omax ) ) @ c_Suc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(250,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__eq__eq_0) ).

thf(1614,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_Suc @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(588,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I14_J_0) ).

thf(2421,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(533,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__SucE_0) ).

thf(2287,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(316,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Omul__0_0) ).

thf(1774,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(939,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ( A @ c_ATP__Linkup_Osko__Nat__Xzero__induct__1__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__induct_0) ).

thf(3232,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ hAPP ) @ hBOOL )
      | ( A @ c_ATP__Linkup_Osko__Nat__Xzero__induct__1__1 @ c_Suc @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(711,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ c_Suc @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power_Opower_Opower__Suc_0) ).

thf(2719,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ c_Suc @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(509,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__0__iff_5) ).

thf(2233,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(255,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__0_0) ).

thf(1625,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(805,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__Suc1_0) ).

thf(2951,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(309,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( B @ c_Lambda_OdB_OAbs ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I3_J_0) ).

thf(1757,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( B @ c_Lambda_OdB_OAbs ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(859,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diffs0__imp__equal_0) ).

thf(3062,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(371,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__one_0) ).

thf(1908,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ c_HOL_Oone__class_Oone @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(410,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__less__iff__disj_2) ).

thf(2003,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(470,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__mult__le__cancel1_0) ).

thf(2142,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Suc @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(797,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__less__imp__less__right_0) ).

thf(2930,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(53,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__iff1_0) ).

thf(1133,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( D @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( E @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( F @ ( E @ ( D @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(569,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OVar ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I5_J_0) ).

thf(2374,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OVar ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(140,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__infE_1) ).

thf(1348,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(539,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supE_1) ).

thf(2302,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(627,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__le__not__le_1) ).

thf(2517,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(296,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__imp__diff__less_0) ).

thf(1725,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(114,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__sum__squares__lt__zero_0) ).

thf(1280,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( C @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(573,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__assoc2_0) ).

thf(2385,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(276,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__antisym__conv_0) ).

thf(1678,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(629,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__mult__distrib_0) ).

thf(2522,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(877,axiom,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gr0I_0) ).

thf(3100,plain,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(893,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ c_InductTermi_OIT )
      | ~ ( G @ ( C @ ( tc_Type_Otype @ ( B @ ( E @ ( F @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( D @ c_InductTermi_OIT )
      | ~ ( B @ ( D @ ( F @ c_Type_Otyping ) ) )
      | ( E @ ( D @ ( C @ c_Lambda_Osubst ) ) @ c_InductTermi_OIT )
      | ( v_T____
       != ( B @ ( A @ c_Type_Otype_OFun ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_MI2_0) ).

thf(3132,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ c_InductTermi_OIT )
      | ~ ( G @ ( C @ ( tc_Type_Otype @ ( B @ ( E @ ( F @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( D @ c_InductTermi_OIT )
      | ~ ( B @ ( D @ ( F @ c_Type_Otyping ) ) )
      | ( E @ ( D @ ( C @ c_Lambda_Osubst ) ) @ c_InductTermi_OIT )
      | ( v_T____
       != ( B @ ( A @ c_Type_Otype_OFun ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(431,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__Suc__eq_2) ).

thf(2050,plain,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(835,axiom,
    tc_nat @ class_Power_Opower,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Power_Opower) ).

thf(3015,plain,
    tc_nat @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(745,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult__mult2_0) ).

thf(2800,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( D @ ( B @ c_Divides_Odiv__class_Odiv ) ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(988,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( E @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Oshift ) ) ) )
      = ( D @ ( C @ ( B @ c_Suc @ ( D @ ( E @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Oshift ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_shift__commute_0) ).

thf(3328,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( E @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Oshift ) ) ) )
      = ( D @ ( C @ ( B @ c_Suc @ ( D @ ( E @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Oshift ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(496,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__mono_0) ).

thf(2201,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(127,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__strict__increasing_0) ).

thf(1314,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(915,axiom,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__0__eq_1) ).

thf(3177,plain,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(723,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__sup__iff_2) ).

thf(2746,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(439,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_OdB_OdB__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( B @ c_Lambda_OdB_OdB__size @ ( A @ c_Lambda_OdB_OdB__size @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osize_I2_J_0) ).

thf(2070,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_OdB_OdB__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ ( B @ c_Lambda_OdB_OdB__size @ ( A @ c_Lambda_OdB_OdB__size @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(586,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__less__Suc__eq_1) ).

thf(2417,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(565,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__iff2_1) ).

thf(2366,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( F @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ c_lessequals ) ) )
      | ( A @ ( F @ ( C @ ( E @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(985,axiom,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__mult,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocomm__monoid__mult) ).

thf(3323,plain,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(418,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__mult__iff_2) ).

thf(2021,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(303,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__le_0) ).

thf(1742,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(978,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = E )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__eq__replicate_1) ).

thf(3309,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = E )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(899,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xupt__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_upt_Oinduct_0) ).

thf(3145,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xupt__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(333,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__nonpos__nonneg_0) ).

thf(1816,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Opordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(201,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel2_2) ).

thf(1495,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(267,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__1_3) ).

thf(1657,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(552,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__increasing__iff_0) ).

thf(2334,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Oone__class_Oone @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(490,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ c_Lambda_Obeta ) )
      | ( B @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Obeta ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_appL_0) ).

thf(2186,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ c_Lambda_Obeta ) )
      | ( B @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Obeta ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(997,axiom,
    ! [A: $i] :
      ( ( A @ c_Suc )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Suc__neq__Zero_0) ).

thf(3347,plain,
    ! [A: $i] :
      ( ( A @ c_Suc )
     != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(963,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__0L_0) ).

thf(3277,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Orderings_Oord__class_Omin ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(229,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__left__mono_0) ).

thf(1561,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(705,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__less__mono2_0) ).

thf(2704,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(810,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__left__idem_0) ).

thf(2961,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__idem__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(289,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Odistrib__sup__le_0) ).

thf(1710,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omin ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(923,axiom,
    tc_nat @ class_OrderedGroup_Ocancel__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocancel__semigroup__add) ).

thf(3196,plain,
    tc_nat @ class_OrderedGroup_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(363,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__double__add__iff__zero__less__single__add_0) ).

thf(1891,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(506,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__max__iff__disj_2) ).

thf(2223,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(743,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_sum__squares__eq__zero__iff_2) ).

thf(2795,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

thf(192,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( C @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ c_Suc @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I27_J_0) ).

thf(1473,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( C @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( B @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ c_Suc @ ( B @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(560,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__increasing_0) ).

thf(2353,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Oone__class_Oone @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(947,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__less__divisor_0) ).

thf(3244,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Omod ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(621,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__right_1) ).

thf(2502,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(60,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv_1) ).

thf(1152,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(614,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__eq__iff_0) ).

thf(2485,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(838,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__commute_0) ).

thf(3020,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(636,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( D @ ( F @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__diff__cong_0) ).

thf(2539,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( A @ ( B @ ( E @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( A @ ( D @ ( F @ c_HOL_Ominus__class_Ominus ) ) @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( E @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
      | ( ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) )
       != ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
      | ~ ( A @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(368,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__less__imp__less_0) ).

thf(1902,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(703,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I12_J_0) ).

thf(2699,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(253,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__inverse2_0) ).

thf(1621,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(568,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__right__mono_0) ).

thf(2372,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(993,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div_H_6) ).

thf(3341,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(1004,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_split__div_0) ).

thf(3362,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(142,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_Obeta ) )
      | ( C @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Obeta ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_appR_0) ).

thf(1352,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_Obeta ) )
      | ( C @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Obeta ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(812,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lessI_0) ).

thf(2966,plain,
    ! [A: $i] : ( tc_nat @ ( A @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(107,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__inverse_0) ).

thf(1268,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(746,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__eq_0) ).

thf(2803,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(217,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel2_1) ).

thf(1532,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(237,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OAbs )
        = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__eq__apps__conv_0) ).

thf(1582,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OAbs )
        = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(523,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mult2__eq_0) ).

thf(2265,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_Divides_Odiv__class_Odiv ) ) @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(144,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__comm__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__strict__left__mono__comm_0) ).

thf(1357,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__comm__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(653,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__linear_0) ).

thf(2583,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(955,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__0__eq__0_0) ).

thf(3259,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(243,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__power_0) ).

thf(1597,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(175,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__less__iff_0) ).

thf(1431,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(834,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less0_0) ).

thf(3013,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(134,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I21_J_0) ).

thf(1331,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(130,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__eq__1__iff_1) ).

thf(1320,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(393,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supE_0) ).

thf(1962,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(594,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__apps__eq__Abs__apps__conv_1) ).

thf(2436,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(346,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__is__add_0) ).

thf(1848,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc )
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(169,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ c_Lambda_Obeta ) )
      | ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Lambda_OdB_OAbs @ c_Lambda_Obeta ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_abs_0) ).

thf(1416,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ c_Lambda_Obeta ) )
      | ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Lambda_OdB_OAbs @ c_Lambda_Obeta ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(667,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__mult__mult1_0) ).

thf(2615,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( D @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(454,axiom,
    ! [A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__0__left_0) ).

thf(2106,plain,
    ! [A: $i] :
      ( ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Oone__class_Oone ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(608,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__left__idem_0) ).

thf(2470,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(26,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__split_0) ).

thf(1069,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(734,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_div__mod__equality_0) ).

thf(2771,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( C @ ( B @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(691,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Suc @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__eq_1) ).

thf(2672,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ c_Suc @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(677,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__increasing_0) ).

thf(2640,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(937,axiom,
    ! [B: $i,A: $i] :
      ~ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Obeta ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_beta__cases_I1_J_0) ).

thf(3227,plain,
    ! [B: $i,A: $i] :
      ~ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Obeta ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(148,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__iff_1) ).

thf(1367,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(386,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mod__self_0) ).

thf(1945,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(973,axiom,
    tc_nat @ class_OrderedGroup_Omonoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Omonoid__add) ).

thf(3299,plain,
    tc_nat @ class_OrderedGroup_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(277,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel2_0) ).

thf(1681,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(74,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__le__iff_1) ).

thf(1184,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(441,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add2_0) ).

thf(2074,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(699,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oinf__sup__distrib1_0) ).

thf(2690,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omin ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omin ) ) @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omin ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(188,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__le__add__iff2_1) ).

thf(1463,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( D @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( E @ ( B @ ( tc_nat @ ( A @ ( D @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( E @ ( B @ ( D @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(826,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ c_Lambda_OdB_OAbs @ ( B @ c_Type_Otyping ) ) )
      | ( A
        = ( C @ ( B @ ( A @ c_Type_Osko__Type__Xtyping__elims__3__2 ) ) @ ( C @ ( B @ ( A @ c_Type_Osko__Type__Xtyping__elims__3__1 ) ) @ c_Type_Otype_OFun ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_typing__elims_I3_J_0) ).

thf(2998,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ c_Lambda_OdB_OAbs @ ( B @ c_Type_Otyping ) ) )
      | ( A
        = ( C @ ( B @ ( A @ c_Type_Osko__Type__Xtyping__elims__3__2 ) ) @ ( C @ ( B @ ( A @ c_Type_Osko__Type__Xtyping__elims__3__1 ) ) @ c_Type_Otype_OFun ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(511,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonpos__nonpos_0) ).

thf(2238,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(13,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__pos__pos_0) ).

thf(1037,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(537,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supI1_0) ).

thf(2297,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(601,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__less__Suc__eq_0) ).

thf(2453,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ c_Suc @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(854,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( tc_Type_Otype @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ( D @ ( C @ c_Type_Otype_OFun ) @ ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs_0) ).

thf(3052,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( tc_Type_Otype @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ( D @ ( C @ c_Type_Otype_OFun ) @ ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(482,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__left__less__imp__less_0) ).

thf(2170,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(559,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__le__cancel__left__neg_1) ).

thf(2351,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__ring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(926,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nitpick_Onat__case__def_0) ).

thf(3200,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(492,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_one__less__mult_0) ).

thf(2191,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Suc @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(331,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__max__iff__disj_0) ).

thf(1812,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(818,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_mult__pos__neg2_0) ).

thf(2980,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(845,axiom,
    tc_nat @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Oorder) ).

thf(3035,plain,
    tc_nat @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(4032,plain,
    $false,
    inference(e,[status(thm)],[1665,1879,1411,1469,2072,2263,3167,2463,2480,1818,1956,2822,2630,2744,1369,1190,2295,3345,2341,2336,3008,2109,1454,2031,2787,2595,1179,2331,2280,1110,2976,3317,2644,2590,2676,3108,1971,1692,1041,1173,2363,1200,2077,3072,1924,1005,2210,1290,1882,3040,3153,1569,1939,2099,2282,2114,1142,1497,3120,2686,3135,2184,2712,1233,1342,2042,1354,1458,1788,3217,3049,1137,1164,1999,2448,3338,2476,2168,1919,3253,2459,1063,1954,2844,1196,2046,1441,3088,1650,3140,2405,1887,2957,1746,3103,2216,3236,2348,3350,2723,2548,1782,2886,1359,2400,2067,1967,2819,1803,2136,1904,1518,1265,1228,1623,1482,1254,2603,3029,2667,2269,2755,1871,2897,2126,2571,2654,1628,3248,1409,1287,3221,2735,2814,2039,3275,1660,1298,2339,3046,2444,1860,1323,3148,1511,2274,2682,3125,1826,1577,3280,1992,3353,1059,3043,1545,1922,1379,1147,1247,1704,2767,2730,1641,2989,2388,1100,3268,2829,1087,1514,1479,1609,1398,3084,1115,2635,3312,1931,2178,1489,1465,1714,3199,3075,2090,2996,2782,2259,3163,1334,2880,2034,2017,2456,2007,1418,2451,1858,2122,1828,1978,1020,3357,1960,3290,3300,2777,3152,2058,1169,1215,2424,1394,2809,1433,2205,1975,2762,2508,2964,3068,1377,2618,2833,3189,1796,3036,2530,1526,2137,2242,1345,2750,1405,1104,1697,3211,2392,2154,1521,3184,2982,1863,2954,3097,3229,3011,2473,2164,3186,2118,2093,2737,2061,1074,1604,2922,3301,3303,2574,1438,3169,3265,2307,3054,2083,1736,2790,1768,1868,2382,1047,1730,1057,1274,2601,2805,2249,1580,1676,1487,3239,2606,2986,2528,3207,3286,1900,1723,3261,1223,1217,2147,3022,1502,1824,1619,2132,2414,3250,1238,2015,1423,3271,2758,2628,2433,3090,1296,1687,1159,1836,1407,2112,1012,1032,1602,2663,1555,2826,3314,3297,3065,2872,1951,1079,2327,1128,2492,1634,1392,3033,1740,1910,2157,1762,2377,2841,2709,1587,1425,2726,3119,3078,3237,1118,3205,3001,2245,1350,3105,1592,1089,1541,1270,1851,1772,1259,3122,1534,1794,1017,2973,1121,3218,1856,1096,1672,1719,2858,2537,3060,3176,3018,1329,2023,1361,1461,2868,2419,2080,1889,1493,2798,1683,3141,2877,1964,1942,2561,3210,2830,1182,2312,3010,1414,2053,2196,2240,1937,2613,3032,1791,2172,2038,1446,2551,1207,2085,2978,2649,1039,3294,1744,3242,2504,2932,2519,1131,1668,2851,2355,2497,2566,1984,2862,3333,2272,2370,1451,2304,1530,1539,2836,1456,1382,2487,2598,3354,2900,1831,1060,1050,2581,2465,1874,3115,2482,1599,2697,3109,3074,3005,3147,1318,1814,1779,3279,2883,1563,2748,2915,2221,1616,1996,2909,1202,2203,1241,2702,1846,1054,1712,3306,3321,1035,2368,1071,2402,2408,1663,1026,1895,1167,2001,1727,2176,2904,2556,2585,1273,2941,1548,3262,3111,1631,3343,2235,2144,1278,1906,1135,1648,1764,3130,1754,2292,2253,2149,3257,2717,1219,2028,1301,2894,1374,1154,1833,3070,1916,3255,2525,2188,2489,2991,1403,2817,1124,2134,2959,2970,1007,3319,3234,2048,2938,2225,1884,3086,1752,1652,1929,3283,2674,2514,2568,1388,3272,1226,2695,1256,1428,2193,1312,3330,1107,2706,1897,2358,1707,3051,2891,1371,3077,2935,1733,2442,2753,2055,1092,2166,1090,2346,2361,2679,2642,2214,2637,2097,2284,2299,2669,2116,3138,1690,1571,1175,1209,2721,1282,2478,3251,1574,1806,2774,3287,2943,2945,3197,1801,1969,3160,1307,2180,2824,1877,2742,1339,1162,2446,3165,3192,1838,1149,1177,2461,2329,1112,2025,2812,2129,2343,2625,2780,2267,2261,1821,1029,3006,2542,2647,1293,2278,1171,1926,2212,2139,2633,2207,2506,2770,2075,1198,1841,1396,2439,1594,1699,2051,2765,1044,1066,1528,2620,3159,2064,1015,3241,3016,2592,1435,3325,3142,2535,2101,1364,2967,2390,1250,3178,1102,1694,2693,2984,1973,1805,2005,1948,2665,1467,2652,1420,2124,1958,3292,1551,1809,2952,1304,3193,2318,2784,1081,2927,1990,3048,2684,1145,2846,2256,2564,1798,1770,2715,2397,3112,2661,1611,1400,3082,3214,2588,3129,3058,1516,3315,1566,2012,2732,1252,1230,2160,1098,2888,3246,1987,3202,1785,3127,2322,1336,2468,2656,3080,1499,1637,2688,1188,1186,3026,2429,1262,1235,2276,2219,2545,2599,3359,3174,2044,1738,3095,2247,1023,3356,2579,1504,2848,1934,2948,2411,2500,1543,1766,2838,2558,2426,1536,1981,1702,2379,1221,1284,1485,2320,1052,1506,1749,3224,1585,1094,2511,1670,3024,3117,2860,2728,1717,2807,1272,3156,2760,2924,2088,3092,1685,1553,2870,3171,1654,1126,3056,3039,2865,2875,1940,1316,1084,1157,1913,2310,1844,2611,2325,2997,1448,3346,1010,1390,2919,1866,2658,2792,2907,1646,1326,2554,3336,1854,1523,2394,1276,1590,1385,1721,2174,1476,1674,2855,1639,2853,2290,2251,2532,1508,2608,3235,2431,3003,1244,1558,2152,2009,1491,3181,3324,1777,3149,2198,2576,1193,1607,2095,2315,1994,2019,2912,2399,1077,2162,2183,1212,2902,2120,2230,1893,2036,1309,2739,2494,1444,2917,2623,1614,2421,2287,1774,3232,2719,2233,1625,2951,1757,3062,1908,2003,2142,2930,1133,2374,1348,2302,2517,1725,1280,2385,1678,2522,3100,3132,2050,3015,2800,3328,2201,1314,3177,2746,2070,2417,2366,3323,2021,1742,3309,3145,1816,1495,1657,2334,2186,3347,3277,1561,2704,2961,1710,3196,1891,2223,2795,1473,2353,3244,2502,1152,2485,3020,2539,1902,2699,1621,2372,3341,3362,1352,2966,1268,2803,1532,1582,2265,1357,2583,3259,1597,1431,3013,1331,1320,1962,2436,1848,1416,2615,2106,2470,1069,2771,2672,2640,3227,1367,1945,3299,1681,1184,2074,2690,1463,2998,2238,1037,2297,2453,3052,2170,2351,3200,2191,1812,2980,3035]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02  % Problem  : LCL796-1 : TPTP v9.3.1. Released v4.1.0.
% 0.00/0.07  % 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
% 0.13/5.41  % Computer : n006.cluster.edu
% 0.13/5.41  % Model    : x86_64 x86_64
% 0.13/5.41  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/5.41  % Memory   : 8046.5625MB
% 0.13/5.41  % OS       : Linux 6.8.0-71-generic
% 0.13/5.41  % CPULimit : 300
% 0.13/5.41  % WCLimit  : 300
% 0.13/5.41  % DateTime : Fri Sep 25 23:34:51 UTC 2026
% 0.13/5.41  % CPUTime  : 
% 0.13/5.41  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
% 0.17/6.00  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 2.37/6.44  % [INFO] 	 Parsing done (430ms). 
% 2.37/6.45  % [INFO] 	 Running in sequential loop mode. 
% 3.26/6.83  % [INFO] 	 eprover registered as external prover. 
% 3.26/6.84  % [INFO] 	 Scanning for conjecture ... 
% 3.93/7.06  % [INFO] 	 Found a conjecture (or negated_conjecture) and 1015 axioms. Running axiom selection ... 
% 5.10/7.38  % [INFO] 	 Axiom selection finished. Selected 1003 axioms (removed 12 axioms). 
% 6.54/7.73  % [INFO] 	 Problem is propositional (TPTP CNF). 
% 6.54/7.76  % [INFO] 	 Type checking passed. 
% 6.54/7.76  % [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 ... 
% 28.77/12.47  % External prover 'e' found a proof!
% 28.77/12.47  % [INFO] 	 Killing All external provers ... 
% 28.77/12.48  % Time passed: 6918ms (effective reasoning time: 6016ms)
% 28.77/12.49  % Axioms used in derivation (1003): cls_mult__eq__1__iff_1, cls_Suc__pred_0, cls_dB_Osplit_0, cls_add__0__left_0, cls_MI1_0, cls_add__pos__nonneg_0, cls_subst__lt_0, cls_mult__div__cancel_0, cls_comm__monoid__add_Opower__mult__distrib_0, cls_dB_Osimps_I6_J_0, clsarity_nat__OrderedGroup_Oab__semigroup__add, cls_not__one__less__zero_0, cls_mult__less__cancel__left__disj_5, cls_less__iff__diff__less__0_0, cls_dB_Orecs_I1_J_0, cls_min__max_Oinf__commute_0, cls_shift__commute_0, cls_add__Suc__shift_0, cls_xt1_I9_J_0, cls_linorder__antisym__conv2_1, cls_one__le__mult__iff_1, cls_power__0_0, cls_nat_Oinduct_0, cls_le__0__eq_1, cls_natgb_Oadd__r0__iff_1, cls_min__max_Ole__infE_0, cls_eq__add__iff2_1, cls_apps__eq__Abs__conv_0, cls_order__le__less__trans_0, cls_min__max_Oless__infI1_0, cls_mult__cancel2_0, cls_add__is__1_1, cls_less__diff__iff_1, cls_mult__zero__right_0, cls_zero__less__one_0, cls_of__nat__add_0, cls_gb__semiring_Omul__0_0, cls_Suc__inject_0, cls_eq__imp__le_0, cls_order__eq__iff_0, cls_div__le__mono_0, cls_comm__semiring__class_Odistrib_0, cls_Suc__mono_0, cls_le__diff__iff_0, cls_gb__semiring_Osemiring__rules_I12_J_0, cls_of__nat__0_0, cls_gb__semiring_Osemiring__rules_I7_J_0, cls_min__max_Osup__inf__distrib1_0, cls_mult__nonneg__nonneg_0, cls_lift_Osimps_I2_J_0, cls_add__less__cancel__left_1, cls_mult__right__mono_0, cls_subst__App_0, cls_order__less__le__trans_0, cls_nat__0__less__mult__iff_1, cls_min__add__distrib__left_0, cls_nat_Osize_I3_J_0, cls_mod__mult__mult1_0, cls_le__add__diff__inverse2_0, cls_Nat_Odiff__diff__eq_0, cls_subst__eq_0, cls_le__diff__conv_1, cls_xt1_I12_J_0, cls_mult__less__cancel__right__disj_0, cls_mult__nonneg__nonpos2_0, cls_min__max_Oinf__sup__distrib1_0, cls_one__is__add_4, cls_min__max_Ole__sup__iff_2, cls_min__max_Oinf__le1_0, cls_class__semiring_Osemiring__rules_I9_J_0, cls_min__less__iff__conj_0, cls_mult__is__0_2, cls_upt_Oinduct_1, cls_nat_Osplit_1, cls_liftn_Osimps_I1_J_1, cls_left__add__mult__distrib_0, cls_dB_Osimps_I8_J_0, cls_sum__squares__eq__zero__iff_0, cls_typing__elims_I3_J_0, cls_subject__reduction_0, cls_Suc__n__not__n_0, cls_mult__less__cancel__left__disj_1, cls_zero__le__one_0, cls_nat__mult__div__cancel__disj_0, cls_nat__add__left__cancel__less_1, cls_power__eq__0__iff_1, cls_termination__basic__simps_I5_J_0, cls_add__increasing2_0, cls_eq__diff__iff_0, cls_gb__semiring_Opwr__pwr_0, cls_min__0R_0, cls_less__antisym_0, cls_less__add__Suc2_0, cls_of__nat__less__iff_0, cls_min__max_Osup__inf__absorb_0, cls_class__semiring_Osemiring__rules_I20_J_0, cls_diff__add__cancel_0, cls_pos__add__strict_0, cls_mod__by__0_0, cls_add__is__1_2, cls_power__inject__exp_0, cls_zero__le__mult__iff_3, cls_linorder__neqE_0, cls_add__strict__right__mono_0, cls_liftn_Osimps_I2_J_0, cls_nat__mod__eq__iff_1, cls_power__increasing_0, cls_one__le__mult__iff_0, clsarity_fun__Orderings_Oorder, cls_not__sum__squares__lt__zero_0, cls_MI2_0, cls_class__semiring_Osemiring__rules_I21_J_0, cls_split__div__lemma_1, cls_add__le__cancel__left_1, cls_uIT_0, cls_class__semiring_Osemiring__rules_I10_J_0, cls_gb__semiring_Osemiring__rules_I3_J_0, cls_min__less__iff__disj_2, cls_not__less__eq_1, cls_appR_0, clsarity_nat__Orderings_Olinorder, cls_zero__induct_1, cls_nat__rec__Suc_0, cls_div__add__self2_0, cls_mod__add__eq_0, cls_termination__basic__simps_I1_J_0, cls_gb__semiring_Opwr__Suc_0, cls_add__is__0_0, clsarity_nat__OrderedGroup_Omonoid__add, cls_Suc__lessI_0, cls_diff__cancel_0, cls_add__imp__eq_0, cls_app__Var__IT_0, cls_nat__eq__add__iff1_0, cls_mod__add__left__eq_0, cls_split__mod_1, cls_less__Suc__eq_2, cls_nat__add__left__cancel__less_0, cls_dB_Osimps_I12_J_0, cls_of__nat__aux_Osimps_I2_J_0, cls_min__less__iff__disj_0, cls_one__is__add_1, cls_mult__nonpos__nonneg_0, cls_add__le__cancel__right_1, cls_gb__semiring_Osemiring__rules_I28_J_0, cls_nat_Osplit__asm_1, cls_power__le__imp__le__exp_0, 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cls_gb__semiring_Osemiring__rules_I22_J_0, cls_nat_Oinject_0, cls_drop__drop_0, cls_even__less__0__iff_1, cls_add__strict__increasing_0, cls_Suc__mult__le__cancel1_1, cls_max__less__iff__conj_1, cls_diff__mult__distrib_0, cls_power__le__imp__le__base_0, cls_power__mult__distrib_0, cls_div__0_0, cls_nat__add__left__cancel__le_1, cls_le__add__diff__inverse_0, cls_min__max_Ole__infI1_0, cls_power__strict__increasing__iff_1, cls_not__less__less__Suc__eq_1, cls_Suc__mult__le__cancel1_0, cls_min__max_Oinf__sup__absorb_0, cls_linorder__less__linear_0, cls_class__semiring_Opwr__Suc_0, clsarity_nat__Power_Opower, cls_power__increasing__iff_1, cls_zero__le__mult__iff_5, cls_nat__diff__add__eq2_0, cls_drop__replicate_0, cls_mult__less__mono2_0, cls_Nat_Oadd__0__right_0, cls_class__semiring_Osemiring__rules_I11_J_0, cls_liftn__lift_0, cls_eq__eqI_0, cls_nat_Osimps_I2_J_0, cls_Zero__neq__Suc_0, cls_less__eq__Suc__le_0, cls_gb__semiring_Osemiring__rules_I18_J_0, cls_Suc__lessD_0, 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cls_min__max_Ole__inf__iff_2, cls_dB_Orecs_I2_J_0, cls_div__mult1__eq_0, cls_typing_OVar_0, cls_mult__1_0, cls_nat__mult__div__cancel1_0, cls_zmod__simps_I2_J_0, cls_mult__less__cancel1_2, cls_mod__mult__self2__is__0_0, cls_add__mult__distrib2_0, cls_abs_0, cls_less__Suc0_0, cls_min__max_Osup__left__commute_0, cls_mod__add__right__eq_0, cls_comm__monoid__add_Omult__left__commute_0, cls_apps__eq__tail__conv_0, cls_diff__0__right_0, cls_xt1_I6_J_0, cls_le__add__right__mono_0, cls_min__max_Ole__iff__sup_1, clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add, cls_gb__semiring_Oadd__a_0, cls_nat__le__add__iff2_1, cls_div__mult__self2_0, cls_min__less__iff__conj_2, cls_diff__Suc__less_0, cls_mult__less__cancel__left__disj_3, cls_zero__less__mult__pos2_0, cls_nat__induct_1, cls_diff__le__self_0, cls_n__less__m__mult__n_0, cls_min__max_Oinf__le2_0, cls_nat__less__le_0, cls_dB_Osplit_1, cls_size__char_0, cls_lift__IT_0, cls_replicate__eq__replicate_2, cls_le__max__iff__disj_2, cls_mod__mult__eq_0, cls_mult__less__cancel__right__disj_4, cls_mult__1__left_0, cls_nat_Osimps_I3_J_0, cls_less__trans__Suc_0, cls_le__Suc__eq_2, cls_min__max_Osup__assoc_0, cls_le__cube_0, cls_min__max_Oinf__absorb2_0, cls_linorder__not__le_0, cls_div__mult__mult2_0, clsarity_nat__Ring__and__Field_Omult__mono1, cls_diff__add__assoc2_0, cls_mult__le__cancel2_0, cls_le__diff__conv2_1, cls_mod__mult__mult2_0, cls_double__add__less__zero__iff__single__less__zero_1, cls_eq__add__iff2_0, cls_diff__less__mono2_0, cls_one__less__mult_0, cls_not__less__eq_0, cls_nat__le__linear_0, cls_trans__less__add1_0, cls_class__semiring_Osemiring__rules_I35_J_0, cls_mult__zero__left_0, cls_plus__nat_Oadd__0_0, cls_nat__rec__0_0, cls_comm__monoid__add_Omult__1_0, cls_liftn_Osimps_I1_J_0, cls_nat__less__le_2, cls_mod__diff__left__eq_0, cls_comm__monoid__add_Opower__mult_0, cls_min__max_Oinf__left__commute_0, cls_apps__preserves__beta_0, cls_order__less__irrefl_0, cls_mod__mult__self3_0, clsarity_nat__Ring__and__Field_Opordered__semiring, cls_add__nonneg__eq__0__iff_2, cls_add__is__0_2, cls_less__diff__iff_0, cls_Abs__eq__apps__conv_0, cls_one__less__power_0, cls_power__Suc__less__one_0, cls_add__le__mono1_0, cls_mult__left__le__imp__le_0, cls_minus__nat_Odiff__0_0, cls_add__is__1_3, cls_le__iff__add_1, cls_class__semiring_Osemiring__rules_I27_J_0, clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add, cls_diff__is__0__eq_1, cls_diff__0__eq__0_0, cls_power__eq__0__iff_0, cls_eq__eqI_1, cls_Suc__n__not__le__n_0, cls_class__semiring_Oadd__c_0, cls_zero__less__diff_1, cls_div__add1__eq_0, cls_mult__pos__pos_0, cls_less__eqI_0, cls_nat__add__left__cancel_0, cls_min__max_Oless__supI2_0, cls_of__nat__aux_Osimps_I1_J_0, cls_mult__strict__right__mono__neg_0, clsarity_nat__Ring__and__Field_Ono__zero__divisors, cls_Suc__leD_0, cls_div__if_0, cls_add__eq__self__zero_0, cls_le__diff__iff_1, cls_add__pos__pos_0, cls_add__mono_0, cls_split__div_H_7, cls_mod__less__divisor_0, cls_n__not__Suc__n_0, cls_le__max__iff__disj_0, cls_min__le__iff__disj_1, cls_sum__squares__gt__zero__iff_0, cls_gb__semiring_Osemiring__rules_I24_J_0, cls_min__diff__distrib__left_0, cls_zmod__simps_I1_J_0, cls_class__semiring_Osemiring__rules_I8_J_0, cls_add__neg__nonpos_0, clsarity_fun__Orderings_Opreorder, cls_min__le__iff__disj_2, cls_power__less__imp__less__base_0, cls_split__mod_5, cls_mult__right__less__imp__less_0, cls_nat__less__add__iff1_1, cls_power__inject__base_0, cls_min__max_Ole__infI_0, cls_nat__le__add__iff1_1, cls_less__SucI_0, cls_nat__mult__le__cancel1_0, cls_nat__0__less__mult__iff_2, cls_not__less0_0, cls_nat__mult__assoc_0, cls_diffs0__imp__equal_0, cls_mod__div__decomp_0, cls_class__semiring_Osemiring__rules_I24_J_0, cls_eq__iff__diff__eq__0_1, cls_min__max_Ole__iff__inf_0, cls_subst__Var__IT_0, cls_shift__gt_0, cls_gb__semiring_Omul__a_0, cls_one__is__add_3, cls_beta__cases_I1_J_0, cls_Var__eq__apps__conv_0, cls_mult__less__cancel__left__disj_2, cls_dB_Osimps_I2_J_0, cls_var__app__type__eq_0, cls_dB_Osimps_I5_J_0, cls_diff__add__inverse_0, cls_gb__semiring_Osemiring__rules_I19_J_0, cls_class__semiring_Opwr__mul_0, cls_nat__mult__commute_0, cls_mult__less__cancel__right__disj_1, cls_add__less__cancel__left_0, cls_zmod__simps_I4_J_0, cls_class__semiring_Osemiring__rules_I25_J_0, cls_min__max_Ole__iff__sup_0, cls_less__max__iff__disj_1, cls_Suc__less__eq_0, cls_zero__le__power_0, cls_max__less__iff__conj_0, clsarity_nat__Ring__and__Field_Osemiring__1, cls_nat__case__Suc_0, cls_order__less__imp__le_0, cls_add__leE_1, cls_dB_Osplit__asm_1, cls_mult__cancel1_0, cls_max__add__distrib__left_0, cls_gb__semiring_Opwr__0_0, cls_abs__typeE_0, cls_le__eqI_1, cls_Lambda_0, cls_class__semiring_Opwr__0_0, cls_Suc__diff__le_0, cls_gb__semiring_Osemiring__rules_I4_J_0, cls_mult__eq__0__iff_1, cls_nat__case__0_0, cls_le__iff__diff__le__0_1, cls_add__gr__0_2, cls_diff__le__mono_0, cls_add__less__le__mono_0, cls_lift_Osimps_I1_J_0, cls_min__max_Ole__supI2_0, cls_gb__semiring_Osemiring__rules_I20_J_0, cls_power__increasing__iff_0, cls_div__mult__self1__is__id_0, cls_add__gr__0_1, cls_min__max_Ole__supE_1, cls_comm__monoid__add_Opower__Suc2_0, cls_mod__eq__0__iff_1, cls_power__strict__increasing__iff_0, cls_upt_Oinduct_0, cls_less__Suc0_1, cls_subst__lift_0, cls_termination__basic__simps_I2_J_0, cls_of__nat__Suc_0, clsarity_nat__Ring__and__Field_Omult__zero, clsarity_nat__OrderedGroup_Ocomm__monoid__add, cls_class__semiring_Omul__pwr_0, cls_mod__mult2__eq_0, cls_gb__semiring_Opwr__mul_0, cls_monoid__add__class_Oadd__0__right_0, cls_of__nat__less__iff_1, cls_mult__idem_0, cls_dB_Osize_I2_J_0, cls_diff__commute_0, cls_dB_Osplit_2, cls_le__add1_0, cls_split__div__lemma_2, cls_mod__if_0, cls_div__mult__mult1__if_0, cls_le__less__Suc__eq_1, cls_add__strict__increasing2_0, cls_le__add__diff_0, cls_semiring__div__class_Omod__div__equality_H_0, cls_le__eq__less__or__eq_0, cls_dB_Osize_I6_J_0, cls_eq__iff__diff__eq__0_0, clsarity_nat__Orderings_Oorder, cls_nat__mult__less__cancel1_0, cls_appL_0, cls_nat__size_0, cls_power__decreasing_0, cls_order__less__le_1, cls_mult__le__0__iff_3, cls_gb__semiring_Osemiring__rules_I11_J_0, cls_min__less__iff__conj_1, cls_xt1_I7_J_0, cls_nat__mult__eq__cancel__disj_0, cls_mult__neg__neg_0, cls_linorder__not__less_1, cls_zero__less__Suc_0, cls_mult__less__cancel__left__neg_0, cls_class__semiring_Osemiring__rules_I4_J_0, cls_not__one__le__zero_0, cls_le__neq__implies__less_0, cls_le__add__iff2_0, cls_nat__less__le_1, cls_mult__less__le__imp__less_0, cls_min__diff_0, cls_Beta_0, cls_mult__le__less__imp__less_0, cls_zero__induct_0, cls_natgb_Oadd__mul__solve_1, cls_min__max_Osup__commute_0, cls_add__le__cancel__left_0, clsarity_nat__Ring__and__Field_Oordered__semiring__strict, clsarity_nat__OrderedGroup_Osemigroup__add, cls_max__Suc__Suc_0, cls_power__Suc__less_0, cls_mod__self_0, cls_gb__semiring_Osemiring__rules_I23_J_0, cls_less__eqI_1, cls_dB_Osimps_I9_J_0, cls_zero__le__imp__of__nat_0, cls_not__less__eq__eq_0, cls_sum__squares__le__zero__iff_0, cls_ex__least__nat__le_1, cls_mod__Suc__eq__Suc__mod_0, cls_add__le__cancel__right_0, cls_gb__semiring_Osemiring__rules_I1_J_0, cls_mult__eq__1__iff_2, cls_mult__less__cancel__left__pos_0, cls_nat__add__commute_0, cls_split__mult__neg__le_1, cls_nat__eq__add__iff2_1, cls_add__diff__inverse_0, cls_power__mono_0, cls_mod__mult__distrib_0, cls_of__nat__less__0__iff_0, cls_Abs__apps__eq__Abs__apps__conv_0, cls_le__refl_0, cls_min__less__iff__disj_1, cls_nat_Oinduct_1, cls_min__0L_0, cls_even__less__0__iff_0, cls_le__SucI_0, cls_gb__semiring_Osemiring__rules_I15_J_0, cls_power__0__left_1, cls_class__semiring_Ogb__semiring__axioms_H_0, cls_class__semiring_Osemiring__rules_I13_J_0, cls_mult__cancel2_1, cls_mult__strict__left__mono__neg_0, cls_class__semiring_Osemiring__rules_I22_J_0, cls_div__if_1, cls_nat_Osize_I1_J_0, cls_zero__le__mult__iff_2
% 28.77/12.49  % No. of inferences in proof: 2009
% 28.77/12.49  % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 6918 ms resp. 6016 ms w/o parsing
% 32.14/13.04  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 32.14/13.05  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------