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

%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : SWW284+1 : TPTP v9.3.1. Released v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39

% Computer : n004.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Sun Sep 27 09:09:49 AM UTC 2026

% Result   : Theorem 52.69s 14.68s
% Output   : Refutation 58.31s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      : 1075
% Syntax   : Number of formulae    : 2152 ( 520 unt;   0 typ;   0 def)
%            Number of atoms       : 5512 (2247 equ;   0 cnn)
%            Maximal formula atoms :   15 (   2 avg)
%            Number of connectives : 30455 ( 472   ~; 114   |; 436   &;26623   @)
%                                         ( 167 <=>;2643  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   7 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  110 ( 108 usr;   8 con; 0-5 aty)
%            Number of variables   : 5284 (   0   ^;5252   !;  32   ?;5284   :)

% Comments : 
%------------------------------------------------------------------------------
thf(v_q_decl,type,
    v_q: $i ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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(c_Groups_Ouminus__class_Ouminus_decl,type,
    c_Groups_Ouminus__class_Ouminus: $i > $i > $i ).

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

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

thf(c_Rings_Odvd__class_Odvd_decl,type,
    c_Rings_Odvd__class_Odvd: $i > $i > $i > $o ).

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

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

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

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

thf(c_fequal_decl,type,
    c_fequal: $i > $i > $i ).

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

thf(c_Rings_Oinverse__class_Oinverse_decl,type,
    c_Rings_Oinverse__class_Oinverse: $i > $i > $i ).

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

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

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

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

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

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

thf(class_Rings_Odivision__ring__inverse__zero_decl,type,
    class_Rings_Odivision__ring__inverse__zero: $i > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(class_Rings_Odvd_decl,type,
    class_Rings_Odvd: $i > $o ).

thf(c_fTrue_decl,type,
    c_fTrue: $i ).

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

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

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

thf(c_Polynomial_Opoly_decl,type,
    c_Polynomial_Opoly: $i > $i > $i ).

thf(c_Polynomial_Omonom_decl,type,
    c_Polynomial_Omonom: $i > $i > $i > $i ).

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

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

thf(c_fFalse_decl,type,
    c_fFalse: $i ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(class_Groups_Osgn__if_decl,type,
    class_Groups_Osgn__if: $i > $o ).

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

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

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

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

thf(c_Polynomial_Opdivmod__rel_decl,type,
    c_Polynomial_Opdivmod__rel: $i > $i > $i > $i > $i > $o ).

thf(c_Polynomial_Oorder_decl,type,
    c_Polynomial_Oorder: $i > $i > $i > $i ).

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

thf(c_Polynomial_Ocoeff_decl,type,
    c_Polynomial_Ocoeff: $i > $i > $i ).

thf(c_Polynomial_Odegree_decl,type,
    c_Polynomial_Odegree: $i > $i > $i ).

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

thf(c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant_decl,type,
    c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant: $i > $i > $i > $o ).

thf(c_Polynomial_OpCons_decl,type,
    c_Polynomial_OpCons: $i > $i > $i > $i ).

thf(c_Polynomial_Osmult_decl,type,
    c_Polynomial_Osmult: $i > $i > $i > $i ).

thf(c_Polynomial_Opoly__gcd_decl,type,
    c_Polynomial_Opoly__gcd: $i > $i > $i > $i ).

thf(c_Polynomial_Osynthetic__div_decl,type,
    c_Polynomial_Osynthetic__div: $i > $i > $i > $i ).

thf(c_Groups_Osgn__class_Osgn_decl,type,
    c_Groups_Osgn__class_Osgn: $i > $i > $i ).

thf(c_Polynomial_OAbs__poly_decl,type,
    c_Polynomial_OAbs__poly: $i > $i > $i ).

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

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

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

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

thf(c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly_decl,type,
    c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly: $i > $i > $i > $i ).

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

thf(c_Polynomial_Opos__poly_decl,type,
    c_Polynomial_Opos__poly: $i > $i > $o ).

thf(c_Polynomial_Opcompose_decl,type,
    c_Polynomial_Opcompose: $i > $i > $i > $i ).

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

thf(c_Nat_Onat_Onat__size_decl,type,
    c_Nat_Onat_Onat__size: $i > $i ).

thf(c_Polynomial_Opoly__rec_decl,type,
    c_Polynomial_Opoly__rec: $i > $i > $i > $i > $i > $i ).

thf(c_If_decl,type,
    c_If: $i > $i > $i > $i > $i ).

thf(c_HOL_Obool_Obool__size_decl,type,
    c_HOL_Obool_Obool__size: $i > $i ).

thf(v_p_decl,type,
    v_p: $i ).

thf(313,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zdiv__trivial) ).

thf(2094,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(775,axiom,
    tc_Nat_Onat @ class_Groups_Oone,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oone) ).

thf(3883,plain,
    tc_Nat_Onat @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(307,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__imp__zdiv__neg__iff) ).

thf(2072,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(671,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B = A )
        | ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Ole__imp__less__or__eq) ).

thf(3434,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B = A )
        | ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(61,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zless__add1__eq) ).

thf(1237,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__diff__eq) ).

thf(3477,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(804,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__1) ).

thf(4015,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(784,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ogroup__add )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__zero) ).

thf(3927,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ogroup__add )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(162,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zless__le) ).

thf(1586,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(141,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__pred) ).

thf(1501,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(862,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ~ ( ? [D: $i] :
              ( ~ ( D @ ( C @ hAPP ) @ hBOOL )
              & ( B
                = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
          | ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
            & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__diff__split__asm) ).

thf(4201,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( ? [D: $i] :
                ( ~ ( D @ ( C @ hAPP ) @ hBOOL )
                & ( B
                  = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
            | ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
              & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
       => ~ ( ? [D: $i] :
                ( ~ ( D @ ( C @ hAPP ) @ hBOOL )
                & ( B
                  = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
            | ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL )
              & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(503,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zminus1__not__zero) ).

thf(2744,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(1059,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__1) ).

thf(4835,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(537,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Omonom ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__monom) ).

thf(2889,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Omonom ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(45,axiom,
    ! [A: $i] :
      ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
        = A )
      & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_DIVISION__BY__ZERO) ).

thf(1190,plain,
    ! [A: $i] :
      ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
        = A )
      & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(922,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
        & ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
            = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zminus2__eq__if) ).

thf(4464,plain,
    ! [A: $i,B: $i] :
      ( ( B
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
        & ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
            = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(399,axiom,
    ! [A: $i,B: $i] :
      ~ ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__add__less2) ).

thf(2341,plain,
    ! [A: $i,B: $i] :
      ~ ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(397,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__Suc) ).

thf(2336,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(295,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel2) ).

thf(2031,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(785,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( B != A )
          & ( D != C ) )
      <=> ( ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
         != ( B @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_crossproduct__noteq) ).

thf(3930,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
           != ( B @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( ( B != A )
            & ( D != C ) ) )
        & ( ( ( B != A )
            & ( D != C ) )
         => ( ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
           != ( B @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(851,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__commute) ).

thf(4169,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(157,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
      <=> ( ( A
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__zmult__eq__1__iff) ).

thf(1559,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( ( A
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
            & ( B
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) )
         => ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) )
        & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
         => ( ( A
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) )
            & ( B
              = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(15,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__semidom )
     => ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__one) ).

thf(1105,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__semidom )
     => ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(435,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__cancel__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__cancel__semiring) ).

thf(2463,plain,
    tc_Nat_Onat @ class_Rings_Oordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(660,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__smult__left) ).

thf(3397,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(515,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) ).

thf(2787,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(443,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ozero__left) ).

thf(2499,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(83,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__inverse__eq) ).

thf(1315,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(770,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__1) ).

thf(3863,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(5,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__add__strict) ).

thf(1083,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(279,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__add__self1) ).

thf(1982,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(41,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__left__commute) ).

thf(1179,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(395,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).

thf(2331,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(1008,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__add) ).

thf(4682,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).

thf(375,axiom,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__0__less__1) ).

thf(2280,plain,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(155,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( D @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__lemma) ).

thf(1554,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( D @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(18,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__1__mult) ).

thf(1110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(605,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__poly__code_I2_J) ).

thf(3172,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(309,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__1__iff) ).

thf(2077,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(725,axiom,
    tc_Nat_Onat @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ono__zero__divisors) ).

thf(3681,plain,
    tc_Nat_Onat @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(163,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc2) ).

thf(1596,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(933,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__mono) ).

thf(4500,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(469,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oord__eq__le__trans) ).

thf(2622,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(413,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__linear) ).

thf(2381,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(123,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mult__less__cancel1) ).

thf(1443,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(291,axiom,
    tc_Int_Oint @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(2014,plain,
    tc_Int_Oint @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(321,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonpos__nonneg) ).

thf(2114,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(980,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ozero__neq__one ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ozero__neq__one) ).

thf(4623,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(969,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1) ).

thf(4592,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(1020,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__vector,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__vector) ).

thf(4714,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__vector,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).

thf(34,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiff__zmod__left) ).

thf(1158,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(448,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__triv__left) ).

thf(2512,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(147,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__less__cancel1) ).

thf(1522,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(749,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_right__minus) ).

thf(3762,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(941,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( D @ c_Polynomial_Opdivmod__rel ) ) ) ) )
      <=> ( ( A
            = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__0__iff) ).

thf(4515,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( ( ( A
              = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( D @ c_Polynomial_Opdivmod__rel ) ) ) ) ) )
        & ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( D @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( ( A
              = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(418,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__div__mult__self) ).

thf(2395,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(542,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( ( C = A )
            & ( D = B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__unique) ).

thf(2908,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( ( C = A )
            & ( D = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(303,axiom,
    tc_Int_Oint @ class_Rings_Oring__1__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1__no__zero__divisors) ).

thf(2062,plain,
    tc_Int_Oint @ class_Rings_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(655,axiom,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__1__left) ).

thf(3377,plain,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(452,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( ( B = A )
        | ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Ole__less) ).

thf(2527,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
            & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B = A )
          | ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
            & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(361,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__Suc__right) ).

thf(2248,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(1053,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) )
          = ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd_Osimps_I2_J) ).

thf(4807,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) )
          = ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).

thf(1015,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__poly__code_I2_J) ).

thf(4697,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).

thf(791,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__dvd__not__less) ).

thf(3959,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(971,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__field,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__field) ).

thf(4603,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__field,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(566,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Nat_Onat_Onat__case ) ) @ hAPP ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__case__0) ).

thf(3012,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Nat_Onat_Onat__case ) ) @ hAPP ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(48,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__lessD1) ).

thf(1205,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(229,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mod) ).

thf(1793,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(662,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat_Oadd__0__right) ).

thf(3402,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(529,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__power__same) ).

thf(2854,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(151,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
        <=> ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mod__iff) ).

thf(1533,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
           => ( B @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) )
          & ( ( B @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
           => ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(657,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__one) ).

thf(3382,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(927,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zminus__inverse2) ).

thf(4479,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(39,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__less__dividend) ).

thf(1173,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(831,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).

thf(4105,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(167,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zmod__equality2) ).

thf(1608,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(744,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oorder__trans) ).

thf(3745,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(464,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__self) ).

thf(2607,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(595,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd_Oleft__commute) ).

thf(3135,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(806,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__mult) ).

thf(4026,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(389,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring) ).

thf(2316,plain,
    tc_Int_Oint @ class_Rings_Ocomm__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(645,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Fields_Ofield )
     => ( ( E
         != ( F @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ ( D @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( A @ ( B @ ( E @ ( F @ c_Rings_Oinverse__class_Oinverse ) @ ( F @ c_Polynomial_Osmult ) ) @ ( C @ ( E @ ( F @ c_Polynomial_Osmult ) ) @ ( D @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__smult__right) ).

thf(3338,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Fields_Ofield )
     => ( ( E
         != ( F @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ ( D @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( A @ ( B @ ( E @ ( F @ c_Rings_Oinverse__class_Oinverse ) @ ( F @ c_Polynomial_Osmult ) ) @ ( C @ ( E @ ( F @ c_Polynomial_Osmult ) ) @ ( D @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(69,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring) ).

thf(1275,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(1069,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__no__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors) ).

thf(4856,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(958,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Osynthetic__div ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_synthetic__div__0) ).

thf(4564,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Osynthetic__div ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(624,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult) ).

thf(3253,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(175,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__self1) ).

thf(1640,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(272,axiom,
    tc_Int_Oint @ class_Int_Oring__char__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Int_Oring__char__0) ).

thf(1954,plain,
    tc_Int_Oint @ class_Int_Oring__char__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(1036,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__0) ).

thf(4762,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).

thf(199,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__simps_I2_J) ).

thf(1705,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(568,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Omonom ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__monom) ).

thf(3017,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Omonom ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(227,axiom,
    ~ ( c_fFalse @ hBOOL ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__fFalse__1) ).

thf(1788,plain,
    ~ ( c_fFalse @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(117,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__neg__pos__less0) ).

thf(1426,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(618,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Ozero__neq__one )
        & ( C @ class_Rings_Ono__zero__divisors )
        & ( C @ class_Rings_Omult__zero )
        & ( C @ class_Power_Opower ) )
     => ( ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__eq__0__iff) ).

thf(3217,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Ozero__neq__one )
        & ( C @ class_Rings_Ono__zero__divisors )
        & ( C @ class_Rings_Omult__zero )
        & ( C @ class_Power_Opower ) )
     => ( ( ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(695,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Polynomial_Omonom ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monom__Suc) ).

thf(3530,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Polynomial_Omonom ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(1061,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oone ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oone) ).

thf(4839,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(652,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__double__add__iff__zero__less__single__add) ).

thf(3367,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(833,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__commute) ).

thf(4110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(945,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__comm__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__comm__semiring) ).

thf(4528,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__comm__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(606,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc )
         != A )
       => ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc )
          = A )
       => ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__Suc) ).

thf(3175,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc )
         != A )
       => ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc )
          = A )
       => ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(991,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__idom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__idom) ).

thf(4650,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(339,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        & ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel1) ).

thf(2168,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          & ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          & ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(817,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__add__distrib) ).

thf(4058,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(90,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_n__less__n__mult__m) ).

thf(1342,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(880,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).

thf(4294,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(55,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__not__refl) ).

thf(1222,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(735,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Osynthetic__div ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Osynthetic__div ) ) @ ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_synthetic__div__pCons) ).

thf(3717,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Osynthetic__div ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Osynthetic__div ) ) @ ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(405,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less__mono2) ).

thf(2360,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(11,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__power) ).

thf(1095,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(814,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Polynomial_Opoly__gcd ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__1__right) ).

thf(4047,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Polynomial_Opoly__gcd ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(354,axiom,
    tc_Int_Oint @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__mult) ).

thf(2216,plain,
    tc_Int_Oint @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(699,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
          = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__equal__iff__equal) ).

thf(3545,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( B = A )
         => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) )
        & ( ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(717,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
     != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_odd__nonzero) ).

thf(3634,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
     != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(688,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__mult__distrib2) ).

thf(3502,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(1006,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ono__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ono__zero__divisors) ).

thf(4679,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ono__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(977,axiom,
    tc_Complex_Ocomplex @ class_Fields_Ofield,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Fields_Ofield) ).

thf(4618,plain,
    tc_Complex_Ocomplex @ class_Fields_Ofield,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(169,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__mono) ).

thf(1613,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(178,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ c_Nat_Onat_Onat__case ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__case__Suc) ).

thf(1645,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ c_Nat_Onat_Onat__case ) ) @ hAPP ) )
      = ( A @ ( B @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(861,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Odiff) ).

thf(4198,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(973,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__0) ).

thf(4607,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(420,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) ).

thf(2400,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(710,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__int__def__symmetric) ).

thf(3597,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(727,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( A @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__smult) ).

thf(3688,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( A @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(259,axiom,
    tc_Int_Oint @ class_Rings_Oordered__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring) ).

thf(1887,plain,
    tc_Int_Oint @ class_Rings_Oordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(898,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__asym) ).

thf(4371,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(824,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) )
        = ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd_Ocommute) ).

thf(4079,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) )
        = ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(633,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__minus__right) ).

thf(3281,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(678,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__add__le__zero__iff__single__add__le__zero) ).

thf(3466,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(998,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__add) ).

thf(4665,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(522,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oring__1__no__zero__divisors )
     => ( ( ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Oone__class_Oone ) )
      <=> ( ( A
            = ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) )
          | ( A
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_square__eq__1__iff) ).

thf(2819,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oring__1__no__zero__divisors )
     => ( ( ( ( A
              = ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) )
            | ( A
              = ( B @ c_Groups_Oone__class_Oone ) ) )
         => ( ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) )
        & ( ( ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) )
         => ( ( A
              = ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) )
            | ( A
              = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(774,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( C = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__right__imp__eq) ).

thf(3880,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( C = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(329,axiom,
    tc_Int_Oint @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ono__zero__divisors) ).

thf(2136,plain,
    tc_Int_Oint @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(145,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__ring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring__1) ).

thf(1518,plain,
    tc_Int_Oint @ class_Rings_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(974,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
          = ( B @ ( A @ ( D @ c_Polynomial_OpCons ) ) ) )
       => ( B
          = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_synthetic__div__unique__lemma) ).

thf(4610,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
          = ( B @ ( A @ ( D @ c_Polynomial_OpCons ) ) ) )
       => ( B
          = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(985,axiom,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__0__0) ).

thf(4637,plain,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(615,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
            = A )
        <=> ( B
            = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__divide__eq__eq) ).

thf(3204,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( B
              = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
           => ( ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
              = A ) )
          & ( ( ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
              = A )
           => ( B
              = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(942,axiom,
    tc_Complex_Ocomplex @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4524,plain,
    tc_Complex_Ocomplex @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(611,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( E @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ ( E @ c_Polynomial_OpCons ) ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__pCons) ).

thf(3195,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( E @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ ( E @ c_Polynomial_OpCons ) ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(632,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pow__divides__eq__nat) ).

thf(3275,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
         => ( C @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( C @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(821,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) ).

thf(4070,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(370,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( C @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zmult2__eq) ).

thf(2269,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( C @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(505,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__diff__conv) ).

thf(2755,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(255,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_field__divide__inverse) ).

thf(1871,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(579,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ( ! [D: $i] :
            ( ( B
              = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
           => ( D @ ( C @ hAPP ) @ hBOOL ) )
        & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__diff__split) ).

thf(3061,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ! [D: $i] :
              ( ( B
                = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
             => ( D @ ( C @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ hAPP ) @ hBOOL )
       => ( ! [D: $i] :
              ( ( B
                = ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
             => ( D @ ( C @ hAPP ) @ hBOOL ) )
          & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(598,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__class_Odistrib) ).

thf(3148,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(71,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__mult__distrib) ).

thf(1279,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I1_J) ).

thf(1832,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(129,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__mono__neg) ).

thf(1462,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(733,axiom,
    tc_Nat_Onat @ class_Groups_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) ).

thf(3713,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(916,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Oadd) ).

thf(4439,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(74,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__zmod) ).

thf(1287,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(280,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Divides_Otransfer__nat__int__function__closures_I2_J) ).

thf(1985,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(912,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__1__iff__1) ).

thf(4422,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
       => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(183,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( B @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__period) ).

thf(1660,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( D @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( ( B @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( B @ ( D @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(78,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc__less) ).

thf(1298,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(398,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__imp__diff__less) ).

thf(2339,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(575,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D @ class_Rings_Odvd )
        & ( D @ class_Rings_Ocomm__ring ) )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ! [E: $i,F: $i] :
            ( ( A @ ( E @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
          <=> ( A @ ( B @ ( F @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inf__period_I3_J) ).

thf(3046,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D @ class_Rings_Odvd )
        & ( D @ class_Rings_Ocomm__ring ) )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ! [E: $i,F: $i] :
            ( ( ( A @ ( B @ ( F @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
             => ( A @ ( E @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) )
            & ( ( A @ ( E @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
             => ( A @ ( B @ ( F @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(432,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__zero__left) ).

thf(2444,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(355,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
              = A )
          <=> ( B
              = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__div__eq__mult) ).

thf(2217,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( ( B
                = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
             => ( ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
                = A ) )
            & ( ( ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
                = A )
             => ( B
                = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(144,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__eq__iff__eq) ).

thf(1511,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( B = A )
         => ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) ) ) )
        & ( ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(372,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_n__less__m__mult__n) ).

thf(2274,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(486,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__add )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(2682,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__add )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(592,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__minus__iff) ).

thf(3125,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(240,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__add) ).

thf(1826,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(746,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( C @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__linear__power) ).

thf(3749,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( C @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(441,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oord__eq__less__trans) ).

thf(2488,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(426,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__trans) ).

thf(2420,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(489,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_plus__nat_Oadd__0) ).

thf(2691,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(586,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__geq) ).

thf(3099,plain,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(1045,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ono__zero__divisors) ).

thf(4788,plain,
    tc_Complex_Ocomplex @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).

thf(31,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__cancel2) ).

thf(1147,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
         => ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(261,axiom,
    tc_Int_Oint @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring__0) ).

thf(1890,plain,
    tc_Int_Oint @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(92,axiom,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__zeroE) ).

thf(1347,plain,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(887,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__0__0) ).

thf(4317,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(762,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A
         != ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
         != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_leading__coeff__neq__0) ).

thf(3835,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A
         != ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
         != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(224,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__cancel__right) ).

thf(1773,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(140,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__less__less__Suc__eq) ).

thf(1494,plain,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( B = A )
         => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(904,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Odiff__right) ).

thf(4390,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(620,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B
            = ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) )
        <=> ( ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__eq__divide__eq) ).

thf(3231,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
              = A )
           => ( B
              = ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) )
          & ( ( B
              = ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) )
           => ( ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
              = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(771,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Oadd__right) ).

thf(3866,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(1001,axiom,
    tc_Complex_Ocomplex @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring__0) ).

thf(4669,plain,
    tc_Complex_Ocomplex @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(497,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
          = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__minus__divide__right) ).

thf(2713,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
          = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(836,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__0) ).

thf(4119,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(104,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring) ).

thf(1379,plain,
    tc_Int_Oint @ class_Rings_Olinordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(654,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide_Oadd) ).

thf(3374,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(658,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__eq__if) ).

thf(3385,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(421,axiom,
    tc_Int_Oint @ class_Groups_Oordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(2403,plain,
    tc_Int_Oint @ class_Groups_Oordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(718,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oantisym__conv) ).

thf(3638,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( ( A = B )
         => ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(7,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__one) ).

thf(1087,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(62,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ? [B: $i] :
          ( A
          = ( B @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gr0__conv__Suc) ).

thf(1247,plain,
    ! [A: $i] :
      ( ( ? [B: $i] :
            ( A
            = ( B @ c_Nat_OSuc ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ? [B: $i] :
            ( A
            = ( B @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(299,axiom,
    ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero )
   != ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__0__neq__1) ).

thf(2049,plain,
    ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero )
   != ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(21,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__increasing) ).

thf(1115,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(546,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( C @ ( A @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__frac__eq) ).

thf(2929,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( C @ ( A @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(679,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) )
      <=> ( C
          = ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__add__iff2) ).

thf(3470,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( C
            = ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) )
        & ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( C
            = ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(266,axiom,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_field__inverse__zero) ).

thf(1931,plain,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(158,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__mult__distrib) ).

thf(1568,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(740,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__divide__eq__iff) ).

thf(3734,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(897,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Opcompose ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opcompose ) ) @ ( A @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pcompose__pCons) ).

thf(4368,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Opcompose ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opcompose ) ) @ ( A @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(843,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
        = ( A @ ( B @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( A @ ( B @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( A @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_offset__poly__pCons) ).

thf(4141,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
        = ( A @ ( B @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( A @ ( B @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( A @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(99,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__SucI) ).

thf(1362,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(274,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__commute) ).

thf(1963,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(1048,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__comm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__comm__monoid__add) ).

thf(4793,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).

thf(936,axiom,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat_Osize_I3_J) ).

thf(4507,plain,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

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

thf(1232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(626,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).

thf(3258,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(523,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Odegree ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__linear__power) ).

thf(2829,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Odegree ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(752,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ? [C: $i] :
          ( B
          = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__eq__0__iff) ).

thf(3788,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ? [C: $i] :
            ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(341,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__less) ).

thf(2178,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(670,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( C @ ( D @ ( E @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__poly__eq) ).

thf(3431,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( C @ ( D @ ( E @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(964,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__div__algebra,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra) ).

thf(4583,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__div__algebra,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(562,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) )
          = ( B @ ( A @ ( D @ c_Polynomial_Omonom ) ) ) )
      <=> ( C = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monom__eq__iff) ).

thf(2996,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( ( C = A )
         => ( ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) )
            = ( B @ ( A @ ( D @ c_Polynomial_Omonom ) ) ) ) )
        & ( ( ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) )
            = ( B @ ( A @ ( D @ c_Polynomial_Omonom ) ) ) )
         => ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(513,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__frac__num) ).

thf(2782,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(602,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__int__def) ).

thf(3163,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(1026,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Olinordered__ab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add) ).

thf(4724,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Olinordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(190,axiom,
    tc_Int_Oint @ class_Groups_Ogroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ogroup__add) ).

thf(1677,plain,
    tc_Int_Oint @ class_Groups_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(511,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__minus__left) ).

thf(2777,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(1054,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__comm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add) ).

thf(4810,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(1014,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__0) ).

thf(4696,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).

thf(358,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          | ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) )
       => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_split__mult__pos__le) ).

thf(2237,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          | ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) )
       => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(113,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__less) ).

thf(1415,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(467,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__neg__div) ).

thf(2616,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(906,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__smult__right) ).

thf(4396,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(184,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__less) ).

thf(1664,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(518,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__cancel1) ).

thf(2794,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B = A ) )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(317,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__self) ).

thf(2105,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(125,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Divides_Osemiring__div )
     => ( ( ( D @ ( E @ ( F @ c_Divides_Odiv__class_Omod ) ) )
          = ( D @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( ( D @ ( B @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
         => ( ( D @ ( B @ ( E @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( C @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__add__cong) ).

thf(1450,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Divides_Osemiring__div )
     => ( ( ( D @ ( E @ ( F @ c_Divides_Odiv__class_Omod ) ) )
          = ( D @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( ( D @ ( B @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
         => ( ( D @ ( B @ ( E @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( C @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(805,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
         != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
      & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zminus1__eq__if) ).

thf(4018,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
         != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
      & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(311,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__diff__diff) ).

thf(2090,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(17,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semidom) ).

thf(1109,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(881,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_OpCons ) ) @ ( B @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__pCons__0) ).

thf(4297,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_OpCons ) ) @ ( B @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(1028,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_offset__poly__0) ).

thf(4733,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).

thf(366,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__add__eq) ).

thf(2259,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(536,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( A
           != ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( ~ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Osgn__class_Osgn ) )
                = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) )
            & ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Osgn__class_Osgn ) )
                = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone ) ) ) ) )
        & ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Osgn__class_Osgn ) )
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__poly__def) ).

thf(2880,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( A
           != ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( ~ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Osgn__class_Osgn ) )
                = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) )
            & ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Osgn__class_Osgn ) )
                = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone ) ) ) ) )
        & ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Osgn__class_Osgn ) )
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(193,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
    <=> ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__neq__iff) ).

thf(1682,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( B != A ) )
      & ( ( B != A )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(28,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zminus__zminus) ).

thf(1136,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(1067,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Oidom )
        & ( B @ class_Int_Oring__char__0 ) )
     => ( ( ( A @ ( B @ c_Polynomial_Opoly ) )
          = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opoly ) ) )
      <=> ( A
          = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__zero) ).

thf(4847,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Oidom )
        & ( B @ class_Int_Oring__char__0 ) )
     => ( ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Polynomial_Opoly ) )
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opoly ) ) ) )
        & ( ( ( A @ ( B @ c_Polynomial_Opoly ) )
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opoly ) ) )
         => ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(845,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__mod__less) ).

thf(4147,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(815,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B
            = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
         => ( ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
            = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__eq__imp) ).

thf(4050,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B
            = ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
         => ( ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(323,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__0) ).

thf(2122,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(106,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
     => ( ( ( C = B )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__less__cases) ).

thf(1383,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
     => ( ( ( C = B )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) )
         => ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(649,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__le__self__iff) ).

thf(3357,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(363,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) ).

thf(2254,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(782,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__diff__right) ).

thf(3923,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(635,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Polynomial_Odegree ) )
          = ( A @ ( C @ c_Polynomial_Odegree ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__add__eq__left) ).

thf(3290,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Polynomial_Odegree ) )
          = ( A @ ( C @ c_Polynomial_Odegree ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(797,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__1) ).

thf(3986,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(867,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc @ c_Nat_Onat_Onat__size )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ c_Nat_Onat_Onat__size @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat_Osize_I2_J) ).

thf(4230,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc @ c_Nat_Onat_Onat__size )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ c_Nat_Onat_Onat__size @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(1063,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ozero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ozero) ).

thf(4842,plain,
    tc_Complex_Ocomplex @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(344,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__power__less__imp__less) ).

thf(2190,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(689,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Opcompose ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__pcompose) ).

thf(3505,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Opcompose ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(650,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__diff__minus) ).

thf(3361,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(120,axiom,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__add) ).

thf(1433,plain,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(316,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__less__SucD) ).

thf(2103,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(553,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__if) ).

thf(2965,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(196,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__diff__1) ).

thf(1697,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(335,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc )
        = ( A @ c_Nat_OSuc ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__inject) ).

thf(2154,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc )
        = ( A @ c_Nat_OSuc ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(825,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__divide__distrib) ).

thf(4082,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(1023,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__ring) ).

thf(4718,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).

thf(1034,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0__right) ).

thf(4751,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).

thf(38,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_fequal ) @ hBOOL )
      | ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__fequal__2) ).

thf(1169,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_fequal ) @ hBOOL )
      | ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(996,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Osgn__if ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Osgn__if) ).

thf(4660,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Osgn__if ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(430,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( A @ c_Nat_OSuc @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__0__Suc) ).

thf(2437,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( A @ c_Nat_OSuc @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(265,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( C @ ( B @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
          = ( C @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mod__cancel) ).

thf(1928,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( C @ ( B @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
          = ( C @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(609,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__diff__left__eq) ).

thf(3189,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(742,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Deriv_Oinverse__diff__inverse) ).

thf(3739,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(376,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zminus__zmod) ).

thf(2281,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(171,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__1) ).

thf(1622,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(409,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__zero) ).

thf(2371,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).

thf(1394,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(601,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oneq__le__trans) ).

thf(3157,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(148,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__decreasing) ).

thf(1526,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(884,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( A @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__1) ).

thf(4307,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( A @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(330,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__power__div__base) ).

thf(2137,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(378,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I2_J) ).

thf(2286,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(811,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) )
          = ( B @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( B @ ( C @ ( D @ c_Groups_Ouminus__class_Ouminus ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
          = ( B @ ( A @ ( D @ c_Groups_Ouminus__class_Ouminus ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__minus__cong) ).

thf(4038,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) )
          = ( B @ ( A @ ( D @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( B @ ( C @ ( D @ c_Groups_Ouminus__class_Ouminus ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
          = ( B @ ( A @ ( D @ c_Groups_Ouminus__class_Ouminus ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(987,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add) ).

thf(4643,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(47,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__not__Zero) ).

thf(1201,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(664,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__pCons__0) ).

thf(3406,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(350,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__div__equality) ).

thf(2205,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(706,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__minus) ).

thf(3582,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(972,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__poly__def) ).

thf(4604,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(708,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) ).

thf(3591,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__offset__poly) ).

thf(3303,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(736,axiom,
    tc_Nat_Onat @ class_Groups_Ozero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ozero) ).

thf(3720,plain,
    tc_Nat_Onat @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(766,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__less) ).

thf(3852,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(608,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__div__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( B @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__mult) ).

thf(3186,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__div__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( B @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(955,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1__no__zero__divisors ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors) ).

thf(4557,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring__1__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(823,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__field )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
            = ( B @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_DERIV__inverse__lemma) ).

thf(4076,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__field )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
            = ( B @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ c_Rings_Oinverse__class_Oinverse ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(1016,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add) ).

thf(4700,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(180,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__right__eq) ).

thf(1651,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(599,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__squared__diff__one__factored) ).

thf(3151,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oring__1 )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(290,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__minus__mult) ).

thf(2011,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(585,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( B @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__dvd1) ).

thf(3097,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( B @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(967,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add) ).

thf(4589,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(983,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) )
          = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      <=> ( B
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monom__eq__0__iff) ).

thf(4629,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(796,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Polynomial_Opos__poly ) )
       => ( ( A @ ( C @ c_Polynomial_Opos__poly ) )
         => ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Polynomial_Opos__poly ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__poly__add) ).

thf(3984,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Polynomial_Opos__poly ) )
       => ( ( A @ ( C @ c_Polynomial_Opos__poly ) )
         => ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Polynomial_Opos__poly ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(604,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__minus__eq) ).

thf(3169,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(298,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zless__mono) ).

thf(2047,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(641,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oidom )
     => ( ~ ( A @ ( B @ c_Polynomial_Opoly ) @ ( B @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant ) ) )
       => ? [C: $i,D: $i] :
            ( ? [E: $i] :
                ( ! [F: $i] :
                    ( ( F @ ( A @ ( B @ c_Polynomial_Opoly ) @ hAPP ) )
                    = ( F @ ( E @ ( D @ ( B @ c_Polynomial_OpCons ) ) @ ( B @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( F @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Opoly ) @ hAPP ) @ ( B @ c_Groups_Oplus__class_Oplus ) ) ) )
                & ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( C @ ( E @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
                  = ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) ) ) )
            & ( C
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( D
             != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__decompose) ).

thf(3318,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Oidom )
     => ( ~ ( A @ ( B @ c_Polynomial_Opoly ) @ ( B @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant ) ) )
       => ? [C: $i,D: $i] :
            ( ? [E: $i] :
                ( ! [F: $i] :
                    ( ( F @ ( A @ ( B @ c_Polynomial_Opoly ) @ hAPP ) )
                    = ( F @ ( E @ ( D @ ( B @ c_Polynomial_OpCons ) ) @ ( B @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( F @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Opoly ) @ hAPP ) @ ( B @ c_Groups_Oplus__class_Oplus ) ) ) )
                & ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( C @ ( E @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
                  = ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) ) ) )
            & ( C
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( D
             != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(200,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__imp__le__left) ).

thf(1708,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(312,axiom,
    tc_Int_Oint @ class_Rings_Oring__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__no__zero__divisors) ).

thf(2093,plain,
    tc_Int_Oint @ class_Rings_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(81,axiom,
    tc_Int_Oint @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__add) ).

thf(1311,plain,
    tc_Int_Oint @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(382,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__Suc1) ).

thf(2296,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(961,axiom,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__zero) ).

thf(4576,plain,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(808,axiom,
    tc_Nat_Onat @ class_Groups_Ocancel__comm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__comm__monoid__add) ).

thf(4031,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

thf(209,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__le__imp__le__exp) ).

thf(1736,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(442,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_left__minus) ).

thf(2496,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(516,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__diff__eq) ).

thf(2790,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(221,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__square) ).

thf(1768,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(541,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Ominus) ).

thf(2905,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(852,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( A @ ( B @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__refl) ).

thf(4172,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( A @ ( B @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(286,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__cancel__left) ).

thf(2000,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(368,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__Suc__right) ).

thf(2264,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(9,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__two) ).

thf(1091,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(519,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__dvd__1__iff__1) ).

thf(2805,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
       => ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(712,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__0__left) ).

thf(3603,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ class_Rings_Osemiring__0 )
        & ( B @ class_Power_Opower ) )
     => ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(146,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__times__mod__eq) ).

thf(1519,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(95,axiom,
    tc_Int_Oint @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Omult__zero) ).

thf(1355,plain,
    tc_Int_Oint @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(63,axiom,
    tc_Int_Oint @ class_Rings_Oordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__semiring) ).

thf(1255,plain,
    tc_Int_Oint @ class_Rings_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(334,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__nonzero__iff__nonzero) ).

thf(2147,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(136,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).

thf(1487,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(713,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Opoly__gcd ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__1__left) ).

thf(3610,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Opoly__gcd ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(1044,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( D @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hAPP ) @ hAPP ) )
          = B )
       => ( ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ ( B @ ( D @ ( A @ c_Polynomial_Opoly__rec ) ) ) ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__rec__0) ).

thf(4785,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( D @ c_Groups_Ozero__class_Ozero @ ( C @ hAPP ) @ hAPP ) @ hAPP ) )
          = B )
       => ( ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ ( B @ ( D @ ( A @ c_Polynomial_Opoly__rec ) ) ) ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(627,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) ).

thf(3261,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(1019,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Int_Oring__char__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Int_Oring__char__0) ).

thf(4712,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Int_Oring__char__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).

thf(570,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oantisym) ).

thf(3022,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(871,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( B
          = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
      <=> ( A
          = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_equation__minus__iff) ).

thf(4252,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( A
            = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
         => ( B
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) )
        & ( ( B
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
         => ( A
            = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(215,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__minus) ).

thf(1751,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(743,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__diff) ).

thf(3742,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

thf(890,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__1__left) ).

thf(4344,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(828,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
          = A )
      <=> ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__equation__iff) ).

thf(4091,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
            = B )
         => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
            = A ) )
        & ( ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
            = A )
         => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(747,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( A @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_synthetic__div__correct_H) ).

thf(3752,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( A @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(621,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__divide__right) ).

thf(3239,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(653,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( D @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd_Oassoc) ).

thf(3371,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( D @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ c_Polynomial_Opoly__gcd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(776,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__imp__not__less) ).

thf(3884,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(600,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__poly__less) ).

thf(3154,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(918,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0) ).

thf(4449,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(170,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__add__left__eq) ).

thf(1619,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(102,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_uminus__dvd__conv_I1_J) ).

thf(1370,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(327,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc__0) ).

thf(2132,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(425,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__less__eq) ).

thf(2414,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(1060,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Osgn__if )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn0) ).

thf(4836,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Osgn__if )
     => ( ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(692,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__group__add )
     => ( ( B @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( A @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__diff__less) ).

thf(3514,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__group__add )
     => ( ( B @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( A @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(726,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Osemiring__0 )
        & ( C @ class_Rings_Odvd ) )
     => ( ? [D: $i] : ( D @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hBOOL )
      <=> ? [D: $i] :
            ( ( D @ ( B @ hAPP ) @ hBOOL )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( D @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_unity__coeff__ex) ).

thf(3682,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Osemiring__0 )
        & ( C @ class_Rings_Odvd ) )
     => ( ( ? [D: $i] :
              ( ( D @ ( B @ hAPP ) @ hBOOL )
              & ( C @ c_Groups_Ozero__class_Ozero @ ( D @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) )
         => ? [D: $i] : ( D @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hBOOL ) )
        & ( ? [D: $i] : ( D @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hBOOL )
         => ? [D: $i] :
              ( ( D @ ( B @ hAPP ) @ hBOOL )
              & ( C @ c_Groups_Ozero__class_Ozero @ ( D @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(292,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zmult1__eq) ).

thf(2015,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(778,axiom,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__mult) ).

thf(3889,plain,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(436,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__if) ).

thf(2464,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(755,axiom,
    tc_Nat_Onat @ class_Rings_Odvd,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Odvd) ).

thf(3810,plain,
    tc_Nat_Onat @ class_Rings_Odvd,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(429,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__neq__Zero) ).

thf(2433,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(612,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__mult__distrib2) ).

thf(3198,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(480,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__left__mono) ).

thf(2663,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(705,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__imp__not__eq2) ).

thf(3578,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(271,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_left__inverse) ).

thf(1951,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(711,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( B @ ( A @ ( D @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
          = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( B
          = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_offset__poly__eq__0__lemma) ).

thf(3600,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( B @ ( A @ ( D @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
          = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( B
          = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(237,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__self2) ).

thf(1819,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(348,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__refl) ).

thf(2200,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(857,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Odegree ) )
            = ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__mult__eq) ).

thf(4187,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Odegree ) )
            = ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(1031,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Oone__class_Oone )
       != ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__neq__zero) ).

thf(4746,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Oone__class_Oone )
       != ( A @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).

thf(50,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc) ).

thf(1210,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(3,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__one__le__zero) ).

thf(1079,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(783,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__semiring) ).

thf(3926,plain,
    tc_Nat_Onat @ class_Rings_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(320,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__mono_H) ).

thf(2112,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( D @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(211,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__two__diff) ).

thf(1740,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(551,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( E
           != ( G @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B
              = ( E @ ( G @ c_Polynomial_Odegree ) @ ( E @ ( G @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( E @ ( G @ c_Polynomial_Odegree ) @ ( C @ ( A @ ( G @ c_Polynomial_OpCons ) ) @ ( G @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( G @ c_Rings_Oinverse__class_Odivide ) ) ) )
           => ( E @ ( B @ ( G @ c_Polynomial_Osmult ) ) @ ( C @ ( A @ ( G @ c_Polynomial_OpCons ) ) @ ( G @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( B @ ( G @ c_Polynomial_OpCons ) ) @ ( E @ ( F @ ( A @ ( G @ c_Polynomial_OpCons ) ) @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__pCons) ).

thf(2958,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( E
           != ( G @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B
              = ( E @ ( G @ c_Polynomial_Odegree ) @ ( E @ ( G @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( E @ ( G @ c_Polynomial_Odegree ) @ ( C @ ( A @ ( G @ c_Polynomial_OpCons ) ) @ ( G @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( G @ c_Rings_Oinverse__class_Odivide ) ) ) )
           => ( E @ ( B @ ( G @ c_Polynomial_Osmult ) ) @ ( C @ ( A @ ( G @ c_Polynomial_OpCons ) ) @ ( G @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( B @ ( G @ c_Polynomial_OpCons ) ) @ ( E @ ( F @ ( A @ ( G @ c_Polynomial_OpCons ) ) @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(116,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__self__is__m) ).

thf(1423,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(630,axiom,
    tc_Nat_Onat @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring__0) ).

thf(3271,plain,
    tc_Nat_Onat @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(603,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oring )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__diff__mult) ).

thf(3166,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oring )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(244,axiom,
    tc_Int_Oint @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Divides_Osemiring__div) ).

thf(1837,plain,
    tc_Int_Oint @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(165,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__swap) ).

thf(1602,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(687,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__minus__left) ).

thf(3499,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(648,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oidom )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
          | ( B
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__cancel__right) ).

thf(3349,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oidom )
     => ( ( ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
            | ( B
              = ( D @ c_Groups_Ozero__class_Ozero ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
            | ( B
              = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(260,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__comm__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__mult__strict__left__mono) ).

thf(1888,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__comm__semiring__strict )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(77,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__mono) ).

thf(1296,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(773,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__antisym) ).

thf(3877,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(908,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero )
          = ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__zero__sym) ).

thf(4402,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ c_Groups_Ozero__class_Ozero )
            = ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) ) ) )
        & ( ( ( B @ c_Groups_Ozero__class_Ozero )
            = ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(697,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
           != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_no__zero__divisors) ).

thf(3535,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
           != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(342,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__cancel1) ).

thf(2181,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
         => ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(352,axiom,
    tc_Int_Oint @ class_Groups_Osgn__if,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Osgn__if) ).

thf(2211,plain,
    tc_Int_Oint @ class_Groups_Osgn__if,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(110,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Divides_Osemiring__div )
     => ( ( ( D @ ( E @ ( F @ c_Divides_Odiv__class_Omod ) ) )
          = ( D @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( ( D @ ( B @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
         => ( ( D @ ( B @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( C @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Divides_Odiv__class_Omod ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__cong) ).

thf(1407,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Divides_Osemiring__div )
     => ( ( ( D @ ( E @ ( F @ c_Divides_Odiv__class_Omod ) ) )
          = ( D @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( ( D @ ( B @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
         => ( ( D @ ( B @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( C @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Divides_Odiv__class_Omod ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(400,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__self2) ).

thf(2344,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(388,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zmod__equality) ).

thf(2313,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(25,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__gt1) ).

thf(1128,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(921,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__left__imp__eq) ).

thf(4461,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(357,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) )
      & ( ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__div__cancel__disj) ).

thf(2228,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) )
      & ( ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(160,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc )
        = ( A @ c_Nat_OSuc ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat_Oinject) ).

thf(1573,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
       => ( ( B @ c_Nat_OSuc )
          = ( A @ c_Nat_OSuc ) ) )
      & ( ( ( B @ c_Nat_OSuc )
          = ( A @ c_Nat_OSuc ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(754,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( B = A )
        | ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel__disj) ).

thf(3799,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B = A )
          | ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( B = A )
          | ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(739,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__zero__right) ).

thf(3731,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Omult__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(8,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__power) ).

thf(1089,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(233,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zero) ).

thf(1804,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(539,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero )
          = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) )
      <=> ( ( B @ c_Groups_Ozero__class_Ozero )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__0__equal__iff__equal) ).

thf(2895,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( ( ( B @ c_Groups_Ozero__class_Ozero )
            = A )
         => ( ( B @ c_Groups_Ozero__class_Ozero )
            = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) )
        & ( ( ( B @ c_Groups_Ozero__class_Ozero )
            = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) )
         => ( ( B @ c_Groups_Ozero__class_Ozero )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(526,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( ( C != B )
         => ( ( B @ ( C @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( D @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( C = B )
         => ( ( B @ ( C @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__monom) ).

thf(2841,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( ( C != B )
         => ( ( B @ ( C @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( D @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( C = B )
         => ( ( B @ ( C @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(832,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Polynomial_Opos__poly ) )
       => ( ( A @ ( C @ c_Polynomial_Opos__poly ) )
         => ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Opos__poly ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__poly__mult) ).

thf(4108,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Polynomial_Opos__poly ) )
       => ( ( A @ ( C @ c_Polynomial_Opos__poly ) )
         => ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Opos__poly ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(57,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__simps_I4_J) ).

thf(1227,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(984,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__group__add) ).

thf(4636,plain,
    tc_Complex_Ocomplex @ class_Groups_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(153,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
    <=> ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__power__eq__Suc__0__iff) ).

thf(1540,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
       => ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(590,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__by__1) ).

thf(3119,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(580,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( B
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( B @ ( C @ c_Polynomial_Odegree ) @ c_Nat_OSuc ) ) )
        & ( ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__pCons__eq__if) ).

thf(3078,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( B
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( B @ ( C @ c_Polynomial_Odegree ) @ c_Nat_OSuc ) ) )
        & ( ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(803,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( E @ c_Polynomial_Omonom ) ) @ ( C @ ( D @ ( E @ c_Polynomial_Omonom ) ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Polynomial_Omonom ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__monom) ).

thf(4012,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( E @ c_Polynomial_Omonom ) ) @ ( C @ ( D @ ( E @ c_Polynomial_Omonom ) ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Polynomial_Omonom ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(672,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__minus__self__iff) ).

thf(3440,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(108,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield__inverse__zero )
     => ( ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( B @ c_Groups_Oone__class_Oone ) )
      <=> ( A
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__eq__1__iff) ).

thf(1397,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield__inverse__zero )
     => ( ( ( A
            = ( B @ c_Groups_Oone__class_Oone ) )
         => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) )
        & ( ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
            = ( B @ c_Groups_Oone__class_Oone ) )
         => ( A
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(793,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) )
          = ( B @ ( C @ c_Polynomial_Odegree ) @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__pCons__eq) ).

thf(3968,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) )
          = ( B @ ( C @ c_Polynomial_Odegree ) @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(98,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( C @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__mono) ).

thf(1360,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Oordered__semiring )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( C @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( B @ ( E @ c_Groups_Ozero__class_Ozero @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(93,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult2__eq) ).

thf(1350,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(219,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiff__zmod__right) ).

thf(1762,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(315,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__assoc) ).

thf(2100,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(67,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__mono__neg) ).

thf(1270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__ring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ c_Groups_Ozero__class_Ozero @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(250,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) ) )
       => ( ( B
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
             != ( C @ c_Groups_Ozero__class_Ozero ) )
           => ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__inverse__eq__imp__eq) ).

thf(1851,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) ) )
       => ( ( B
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
             != ( C @ c_Groups_Ozero__class_Ozero ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(223,axiom,
    tc_Int_Oint @ class_Groups_Ozero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ozero) ).

thf(1772,plain,
    tc_Int_Oint @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(591,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__divide) ).

thf(3122,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(560,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__unique) ).

thf(2990,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(939,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__field,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__field) ).

thf(4512,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__field,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(487,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__num__frac) ).

thf(2685,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(1046,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1) ).

thf(4789,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).

thf(949,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring__1__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors) ).

thf(4544,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(252,axiom,
    tc_Int_Oint @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__add) ).

thf(1856,plain,
    tc_Int_Oint @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(490,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = C ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power_Opower_Opower__0) ).

thf(2694,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( D @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(943,axiom,
    tc_Complex_Ocomplex @ class_Rings_Odvd,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Odvd) ).

thf(4525,plain,
    tc_Complex_Ocomplex @ class_Rings_Odvd,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(1013,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Ozero__class_Ozero )
       != ( A @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__neq__one) ).

thf(4693,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ozero__neq__one )
     => ( ( A @ c_Groups_Ozero__class_Ozero )
       != ( A @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).

thf(1066,axiom,
    ( ( c_fTrue @ c_HOL_Obool_Obool__size )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_bool_Osize_I1_J) ).

thf(4845,plain,
    ( ( c_fTrue @ c_HOL_Obool_Obool__size )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(447,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide_Odiff) ).

thf(2509,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(587,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__imp__le) ).

thf(3102,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(868,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_leading__coeff__0__iff) ).

thf(4233,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(187,axiom,
    tc_Int_Oint @ class_Rings_Odvd,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Odvd) ).

thf(1672,plain,
    tc_Int_Oint @ class_Rings_Odvd,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(730,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__1) ).

thf(3704,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(501,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( D @ ( C @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) )
         => ( ( D @ ( B @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ) )
     => ( ( ( C @ ( tc_Complex_Ocomplex @ c_Polynomial_Odegree ) )
          = A )
       => ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( B @ ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nullstellensatz__lemma) ).

thf(2724,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( ( D @ ( C @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) )
         => ( ( D @ ( B @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ) )
     => ( ( ( C @ ( tc_Complex_Ocomplex @ c_Polynomial_Odegree ) )
          = A )
       => ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( B @ ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(204,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pow__divides__pow__int) ).

thf(1719,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(866,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__diff__inverse) ).

thf(4227,plain,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(434,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( C @ ( D @ ( E @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__poly__eq) ).

thf(2460,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( C @ ( D @ ( E @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(87,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__mult2) ).

thf(1329,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(392,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_division__ring__inverse__diff) ).

thf(2323,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( A @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(76,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__cancel__left) ).

thf(1292,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(42,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Oone__class_Oone ) )
       => ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__unique) ).

thf(1182,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Oone__class_Oone ) )
       => ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(734,axiom,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra__1 )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__one) ).

thf(3714,plain,
    ! [A: $i] :
      ( ( A @ class_RealVector_Oreal__normed__algebra__1 )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(1033,axiom,
    tc_Complex_Ocomplex @ class_Fields_Ofield__inverse__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Fields_Ofield__inverse__zero) ).

thf(4750,plain,
    tc_Complex_Ocomplex @ class_Fields_Ofield__inverse__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(950,axiom,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Oring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Divides_Oring__div) ).

thf(4545,plain,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(946,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) )
          = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__eq__0__iff) ).

thf(4530,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(1005,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__poly__code_I1_J) ).

thf(4676,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).

thf(879,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) ).

thf(4291,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(597,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__left__cancel) ).

thf(3141,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B = A )
         => ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) )
        & ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(535,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).

thf(2877,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(584,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add_Ocomm__neutral) ).

thf(3094,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(850,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__greatest) ).

thf(4167,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(859,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( ( D @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) )
              = ( C @ ( A @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) )
          <=> ( ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
              = ( D @ ( A @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_frac__eq__eq) ).

thf(4191,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( ( ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
                = ( D @ ( A @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
             => ( ( D @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) )
                = ( C @ ( A @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) ) )
            & ( ( ( D @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) )
                = ( C @ ( A @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) )
             => ( ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
                = ( D @ ( A @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(940,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring) ).

thf(4513,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(197,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__less__mult) ).

thf(1700,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(466,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__self) ).

thf(2613,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(572,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__1__no__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
         != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_field__power__not__zero) ).

thf(3032,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__1__no__zero__divisors )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
         != ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(235,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__self4) ).

thf(1810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(1071,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ~ ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_Opos__poly ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__pos__poly__0) ).

thf(4861,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ~ ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_Opos__poly ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(990,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oidom) ).

thf(4648,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(816,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Polynomial_Opos__poly ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__poly__def) ).

thf(4054,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Polynomial_Opos__poly ) )
         => ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Polynomial_Opos__poly ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(318,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__div__cancel) ).

thf(2108,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(296,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__cancel__less) ).

thf(2038,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(455,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__div__minus__left) ).

thf(2551,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(49,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__self2__is__0) ).

thf(1207,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(738,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oone )
     => ( ( ( B @ c_Groups_Oone__class_Oone )
          = A )
      <=> ( A
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__reorient) ).

thf(3723,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oone )
     => ( ( ( A
            = ( B @ c_Groups_Oone__class_Oone ) )
         => ( ( B @ c_Groups_Oone__class_Oone )
            = A ) )
        & ( ( ( B @ c_Groups_Oone__class_Oone )
            = A )
         => ( A
            = ( B @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(1074,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring) ).

thf(4867,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(701,axiom,
    ! [A: $i,B: $i] :
      ( ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__1__eq__mult__iff) ).

thf(3555,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) )
       => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(177,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__idom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__idom) ).

thf(1644,plain,
    tc_Int_Oint @ class_Rings_Olinordered__idom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(1056,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( ( ( C
             != ( D @ c_Groups_Ozero__class_Ozero ) )
           => ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
          & ( ( C
              = ( D @ c_Groups_Ozero__class_Ozero ) )
           => ( A
              = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__dvd__iff) ).

thf(4814,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( ( ( ( C
               != ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
            & ( ( C
                = ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( A
                = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) )
         => ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( ( C
               != ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
            & ( ( C
                = ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( A
                = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).

thf(437,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) ).

thf(2472,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(979,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__comm__semiring__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict) ).

thf(4621,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__comm__semiring__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(855,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__add__less__zero__iff__single__add__less__zero) ).

thf(4180,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(390,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zmult1__eq) ).

thf(2317,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(1029,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) )
          = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pCons__eq__0__iff) ).

thf(4736,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).

thf(622,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B = A )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oord__le__eq__trans) ).

thf(3242,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B = A )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(902,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Polynomial_Osmult ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd_Osimps_I1_J) ).

thf(4384,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Opoly__gcd ) ) )
        = ( A @ ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Polynomial_Osmult ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(384,axiom,
    tc_Int_Oint @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Power_Opower) ).

thf(2304,plain,
    tc_Int_Oint @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(127,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) )
      = ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zminus__zadd__distrib) ).

thf(1456,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) )
      = ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(528,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__minus__divide__divide) ).

thf(2851,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(928,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__0) ).

thf(4482,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(947,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra) ).

thf(4540,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(256,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__mult2) ).

thf(1874,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(278,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).

thf(1979,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(331,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__less__add2) ).

thf(2140,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(173,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__zero__less__power__iff) ).

thf(1627,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(239,axiom,
    tc_Nat_Onat @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Divides_Osemiring__div) ).

thf(1825,plain,
    tc_Nat_Onat @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(85,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__div__less__self) ).

thf(1324,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(20,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__semidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semidom) ).

thf(1114,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(547,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__divide__eq__iff) ).

thf(2932,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(683,axiom,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__mult) ).

thf(3486,plain,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(324,axiom,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__Suc) ).

thf(2125,plain,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(13,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__increasing__iff) ).

thf(1099,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) )
          & ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(445,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__comm__semiring__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) ).

thf(2505,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(758,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Odiff) ).

thf(3819,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(520,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__mult) ).

thf(2813,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(891,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Polynomial_Odegree ) )
        = ( A @ ( B @ c_Polynomial_Odegree ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__minus) ).

thf(4347,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Polynomial_Odegree ) )
        = ( A @ ( B @ c_Polynomial_Odegree ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(119,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Zero__neq__Suc) ).

thf(1429,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(665,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__0__right) ).

thf(3409,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Ominus__class_Ominus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(300,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat_Osimps_I2_J) ).

thf(2052,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(164,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
         != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__imp__inverse__nonzero) ).

thf(1599,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
         != ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(491,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__distrib2) ).

thf(2697,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(651,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__minus__cancel) ).

thf(3364,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(781,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).

thf(3920,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(371,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mono) ).

thf(2272,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(613,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__1) ).

thf(3201,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(218,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( C @ ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( C @ ( A @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__add) ).

thf(1759,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( C @ ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( C @ ( A @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(748,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__eq__0__iff) ).

thf(3755,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(84,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__bin__lemma) ).

thf(1318,plain,
    ! [A: $i,B: $i] :
      ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(545,axiom,
    ! [A: $i,B: $i] :
      ( ( B
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
     => ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__self__implies__10) ).

thf(2926,plain,
    ! [A: $i,B: $i] :
      ( ( B
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
     => ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(988,axiom,
    tc_Complex_Ocomplex @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Power_Opower) ).

thf(4644,plain,
    tc_Complex_Ocomplex @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(506,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__eq__plus1__left) ).

thf(2761,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(981,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__add) ).

thf(4625,plain,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(607,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__minus) ).

thf(3183,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(926,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( C @ ( B @ hAPP ) )
          = ( C @ ( A @ hAPP ) ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ext) ).

thf(4476,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( C @ ( B @ hAPP ) )
          = ( C @ ( A @ hAPP ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(767,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Power_Opower )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__0) ).

thf(3856,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Power_Opower )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(638,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__1__right) ).

thf(3306,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(308,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__1) ).

thf(2076,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(800,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) ).

thf(3999,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(131,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__less__eq) ).

thf(1466,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(43,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__mono) ).

thf(1185,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(723,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( C @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__synthetic__div) ).

thf(3675,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( C @ c_Polynomial_Odegree ) )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(473,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__reduce) ).

thf(2634,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(121,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_uminus__dvd__conv_I2_J) ).

thf(1434,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(253,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ? [C: $i] :
            ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
            & ( B
              = ( C @ c_Nat_OSuc ) ) )
        | ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__Suc__eq__0__disj) ).

thf(1857,plain,
    ! [A: $i,B: $i] :
      ( ( ( ? [C: $i] :
              ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
              & ( B
                = ( C @ c_Nat_OSuc ) ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ? [C: $i] :
              ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
              & ( B
                = ( C @ c_Nat_OSuc ) ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(856,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__divide__distrib) ).

thf(4184,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( A @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(959,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring__1) ).

thf(4567,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(351,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).

thf(2208,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(195,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__mod__bound) ).

thf(1695,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(1037,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Osmult ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__0__right) ).

thf(4765,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Osmult ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(44,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__div__mult) ).

thf(1187,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(431,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__less__nonneg) ).

thf(2440,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(423,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__simps_I1_J) ).

thf(2408,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(517,axiom,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__add) ).

thf(2793,plain,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(690,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A @ ( B @ c_Polynomial_Ocoeff ) @ ( B @ c_Polynomial_OAbs__poly ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__inverse) ).

thf(3508,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A @ ( B @ c_Polynomial_Ocoeff ) @ ( B @ c_Polynomial_OAbs__poly ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(37,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__imp__less__left) ).

thf(1167,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(476,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__poly__0__right) ).

thf(2645,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(669,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Opos__poly ) )
      <=> ( ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
            & ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
          | ( A @ ( C @ c_Polynomial_Opos__poly ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__poly__pCons) ).

thf(3422,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( ( ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
              & ( A
                = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
            | ( A @ ( C @ c_Polynomial_Opos__poly ) ) )
         => ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Opos__poly ) ) )
        & ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Opos__poly ) )
         => ( ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
              & ( A
                = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
            | ( A @ ( C @ c_Polynomial_Opos__poly ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(179,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__neqE__linordered__idom) ).

thf(1648,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B != A )
       => ( ~ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(484,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ ( E @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ ( E @ c_Polynomial_OpCons ) ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__pCons) ).

thf(2677,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ ( E @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ ( E @ c_Polynomial_OpCons ) ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(975,axiom,
    ! [A: $i] :
      ( ( A @ ( v_q @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
      = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(4613,plain,
    ! [A: $i] :
      ( ( A @ ( v_q @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
      = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(779,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Polynomial_Opdivmod__rel ) ) ) ) )
      <=> ( ( ( C
             != ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
           => ( ( C @ ( E @ c_Polynomial_Odegree ) @ ( A @ ( E @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
              | ( A
                = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) )
          & ( ( C
              = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
           => ( B
              = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
          & ( D
            = ( A @ ( C @ ( B @ ( E @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__def) ).

thf(3890,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( ( ( ( C
               != ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
             => ( ( C @ ( E @ c_Polynomial_Odegree ) @ ( A @ ( E @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
                | ( A
                  = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) )
            & ( ( C
                = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
             => ( B
                = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
            & ( D
              = ( A @ ( C @ ( B @ ( E @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) )
         => ( A @ ( B @ ( C @ ( D @ ( E @ c_Polynomial_Opdivmod__rel ) ) ) ) ) )
        & ( ( A @ ( B @ ( C @ ( D @ ( E @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( ( ( C
               != ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
             => ( ( C @ ( E @ c_Polynomial_Odegree ) @ ( A @ ( E @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
                | ( A
                  = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) )
            & ( ( C
                = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
             => ( B
                = ( E @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
            & ( D
              = ( A @ ( C @ ( B @ ( E @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(135,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat_Osimps_I3_J) ).

thf(1483,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(905,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__nat_Odiff__0) ).

thf(4393,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(764,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__divide__left) ).

thf(3839,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(874,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( C @ ( A @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__frac__eq) ).

thf(4269,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( C @ ( A @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( A @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(896,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pow__divides__pow__nat) ).

thf(4365,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(731,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__eq__plus1) ).

thf(3707,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
      = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(207,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( ( C
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) )
        & ( ( C
            = ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__mult1__if) ).

thf(1727,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( ( C
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) )
        & ( ( C
            = ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(923,axiom,
    ! [A: $i] : ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gcd__lcm__complete__lattice__nat_Otop__greatest) ).

thf(4470,plain,
    ! [A: $i] : ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(216,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__by__0) ).

thf(1754,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(835,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__mod__minus__right) ).

thf(4116,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(14,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__less__imp__less__exp) ).

thf(1103,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(970,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Polynomial_Opdivmod__rel ) ) ) ) )
      <=> ( ( A = C )
          & ( B
            = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__by__0__iff) ).

thf(4594,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( ( ( A = C )
            & ( B
              = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Polynomial_Opdivmod__rel ) ) ) ) ) )
        & ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ ( D @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( ( A = C )
            & ( B
              = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(53,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__zmod__imp__zdvd) ).

thf(1219,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(910,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( B
          = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
      <=> ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__neg__iff__add__eq__0) ).

thf(4412,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) )
        & ( ( B
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(550,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
    <=> ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
        | ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__is__1) ).

thf(2941,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
       => ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(451,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__right) ).

thf(2524,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(411,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__div__cancel) ).

thf(2376,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(32,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__0) ).

thf(1154,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(997,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0__left) ).

thf(4662,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__add )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(889,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__1) ).

thf(4341,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(751,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
          = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) )
      <=> ( ( ( A
              = ( D @ c_Groups_Ozero__class_Ozero ) )
           => ( C
              = ( D @ c_Groups_Ozero__class_Ozero ) ) )
          & ( ( A
             != ( D @ c_Groups_Ozero__class_Ozero ) )
           => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
              = B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__divide__eq) ).

thf(3768,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( ( ( ( A
                = ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( C
                = ( D @ c_Groups_Ozero__class_Ozero ) ) )
            & ( ( A
               != ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
                = B ) ) )
         => ( C
            = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) )
        & ( ( C
            = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) )
         => ( ( ( A
                = ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( C
                = ( D @ c_Groups_Ozero__class_Ozero ) ) )
            & ( ( A
               != ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
                = B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(549,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( C = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__unique__mod) ).

thf(2938,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(356,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zero) ).

thf(2225,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(756,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__0) ).

thf(3811,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(694,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__is__0) ).

thf(3519,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(343,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__left__mono) ).

thf(2188,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(449,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant ) ) )
    <=> ! [D: $i,E: $i] :
          ( ( D @ ( A @ hAPP ) )
          = ( E @ ( A @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_constant__def) ).

thf(2514,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i,E: $i] :
            ( ( D @ ( A @ hAPP ) )
            = ( E @ ( A @ hAPP ) ) )
       => ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant ) ) ) )
      & ( ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant ) ) )
       => ! [D: $i,E: $i] :
            ( ( D @ ( A @ hAPP ) )
            = ( E @ ( A @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(809,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__1__right) ).

thf(4032,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(676,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( ( A
            = ( C @ c_Groups_Oone__class_Oone ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__power) ).

thf(3458,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( ( A
            = ( C @ c_Groups_Oone__class_Oone ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(498,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__pCons) ).

thf(2716,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(23,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__gt1__lemma) ).

thf(1124,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(64,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__left__mono) ).

thf(1256,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(935,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__add) ).

thf(4505,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(812,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__greater) ).

thf(4041,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(844,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__imp__mod__0) ).

thf(4144,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(876,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__zero) ).

thf(4283,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(625,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__div__minus__right) ).

thf(3255,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(417,axiom,
    ! [A: $i,B: $i] : ( B @ c_Nat_OSuc @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less__Suc) ).

thf(2393,plain,
    ! [A: $i,B: $i] : ( B @ c_Nat_OSuc @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(582,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__poly__gcd__iff) ).

thf(3086,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( ( ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
            & ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
         => ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
            & ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(631,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( C @ c_Polynomial_Odegree ) )
        = ( B @ ( C @ c_Polynomial_Odegree ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__offset__poly) ).

thf(3272,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( C @ c_Polynomial_Odegree ) )
        = ( B @ ( C @ c_Polynomial_Odegree ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(837,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Polynomial_Odegree ) )
          = ( A @ ( C @ c_Polynomial_Odegree ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__add__eq__right) ).

thf(4122,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Polynomial_Odegree ) )
          = ( A @ ( C @ c_Polynomial_Odegree ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(230,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__commute) ).

thf(1795,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(419,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__not__zless) ).

thf(2398,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ~ ( B @ ( A @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(968,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring) ).

thf(4590,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(1039,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ozero__neq__one) ).

thf(4769,plain,
    tc_Complex_Ocomplex @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(118,axiom,
    tc_Int_Oint @ class_Groups_Ocancel__comm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__comm__monoid__add) ).

thf(1428,plain,
    tc_Int_Oint @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(929,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Polynomial_OpCons ) ) @ ( C @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__pCons__right) ).

thf(4485,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Polynomial_OpCons ) ) @ ( C @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(186,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__self) ).

thf(1669,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(393,axiom,
    ! [A: $i,B: $i] :
      ~ ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__add__less1) ).

thf(2326,plain,
    ! [A: $i,B: $i] :
      ~ ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(381,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__zero__imp__zero) ).

thf(2293,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(82,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mod__trivial) ).

thf(1312,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(863,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( C @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__trans) ).

thf(4218,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( C @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(548,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide_Ominus) ).

thf(2935,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(16,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__one__less__zero) ).

thf(1107,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ~ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(709,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( B @ c_Groups_Osgn__class_Osgn ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__sgn) ).

thf(3594,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) @ ( B @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( B @ c_Groups_Osgn__class_Osgn ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(88,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
        = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
    <=> ? [C: $i] :
          ( B
          = ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__eq__0__iff) ).

thf(1332,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( B
            = ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ? [C: $i] :
            ( B
            = ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(684,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__comm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__comm__semiring) ).

thf(3487,plain,
    tc_Nat_Onat @ class_Rings_Oordered__comm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(729,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = B )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__eq__self__zero) ).

thf(3700,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = B )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(65,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__power__nat__eq) ).

thf(1258,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(944,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__0) ).

thf(4526,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(29,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gr__implies__not0) ).

thf(1139,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(753,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Ominus) ).

thf(3796,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(1050,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
        <=> ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__smult__iff) ).

thf(4796,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
           => ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
          & ( ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
           => ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(386,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).

thf(2308,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(989,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__0) ).

thf(4645,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(829,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( D = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__unique__div) ).

thf(4099,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G @ class_Fields_Ofield )
     => ( ( C @ ( D @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( E @ ( F @ ( G @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( D = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(643,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__diff) ).

thf(3330,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(761,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).

thf(3832,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(220,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zpower__zmod) ).

thf(1765,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(1068,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4854,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(264,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( C @ hAPP ) @ hBOOL )
    <=> ( ( ( A
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ! [D: $i,E: $i] :
              ( ( A @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( B
                  = ( E @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
               => ( E @ ( C @ hAPP ) @ hBOOL ) ) ) )
        & ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( B @ ( C @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_split__mod) ).

thf(1897,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ! [D: $i,E: $i] :
                ( ( A @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
               => ( ( B
                    = ( E @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
                 => ( E @ ( C @ hAPP ) @ hBOOL ) ) ) )
          & ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ( B @ ( C @ hAPP ) @ hBOOL ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( C @ hAPP ) @ hBOOL ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( C @ hAPP ) @ hBOOL )
       => ( ( ( A
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ! [D: $i,E: $i] :
                ( ( A @ ( E @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
               => ( ( B
                    = ( E @ ( D @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
                 => ( E @ ( C @ hAPP ) @ hBOOL ) ) ) )
          & ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
           => ( B @ ( C @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(986,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__poly__code_I1_J) ).

thf(4640,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(428,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__Suc0) ).

thf(2425,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(667,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__smult) ).

thf(3416,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(583,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Ozero) ).

thf(3091,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(130,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__trans__Suc) ).

thf(1464,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(914,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__power) ).

thf(4433,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(96,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__0) ).

thf(1356,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(387,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__strict__right__mono) ).

thf(2311,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(1047,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
        = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_offset__poly__single) ).

thf(4790,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
        = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).

thf(799,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__cancel2) ).

thf(3996,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(475,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__add) ).

thf(2642,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__add )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(414,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
        = ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__divide) ).

thf(2383,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
        = ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(510,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__divide__divide) ).

thf(2774,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(668,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__group__add )
     => ( ( B @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Omonom ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__monom) ).

thf(3419,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__group__add )
     => ( ( B @ ( A @ ( D @ c_Polynomial_Omonom ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Omonom ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(314,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__eq__less) ).

thf(2097,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(139,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__imp__less__right) ).

thf(1492,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(732,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat_Osize_I4_J) ).

thf(3710,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(882,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = A )
         => ( B
            = ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__divide__imp) ).

thf(4300,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = A )
         => ( B
            = ( C @ ( A @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(900,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) )
    <=> ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__eq__dvd__iff) ).

thf(4374,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(347,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__Suc__eq__Suc__mod) ).

thf(2197,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(40,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          | ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) )
       => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_split__mult__neg__le) ).

thf(1175,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          | ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) )
       => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(1022,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__semiring) ).

thf(4716,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).

thf(724,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__diff) ).

thf(3678,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(932,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__triv__right) ).

thf(4498,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(72,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__eq__imp__eq) ).

thf(1282,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(263,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__mult) ).

thf(1894,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Omonoid__mult )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(559,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).

thf(2987,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(574,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oidom )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
          | ( C
            = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__cancel__left) ).

thf(3038,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oidom )
     => ( ( ( ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
            | ( C
              = ( D @ c_Groups_Ozero__class_Ozero ) ) )
         => ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
            | ( C
              = ( D @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(427,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__div__mult__right) ).

thf(2422,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(22,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__inject__exp) ).

thf(1117,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( ( B = A )
           => ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) )
          & ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
              = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(364,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring) ).

thf(2257,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(377,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__comm__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__mult__left__mono) ).

thf(2284,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__comm__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(205,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( C @ ( A @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
          <=> ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__dvd__div) ).

thf(1722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
             => ( C @ ( A @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) )
            & ( ( C @ ( A @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
             => ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(1052,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(4805,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(30,axiom,
    ! [A: $i] :
      ( ( A @ class_Power_Opower )
     => ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ c_Groups_Otimes__class_Otimes @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Power_Opower_Opower ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__power__power) ).

thf(1144,plain,
    ! [A: $i] :
      ( ( A @ class_Power_Opower )
     => ( ( A @ c_Power_Opower__class_Opower )
        = ( A @ c_Groups_Otimes__class_Otimes @ ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Power_Opower_Opower ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(966,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__smult__right) ).

thf(4586,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(322,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Divides_Osemiring__div )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( D @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
           => ( ( A @ ( C @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
             => ( ( ( D @ ( B @ ( E @ c_Divides_Odiv__class_Odiv ) ) )
                  = ( C @ ( A @ ( E @ c_Divides_Odiv__class_Odiv ) ) ) )
              <=> ( ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
                  = ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__div__div__eq__mult) ).

thf(2116,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Divides_Osemiring__div )
     => ( ( D
         != ( E @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C
           != ( E @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( D @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
           => ( ( A @ ( C @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
             => ( ( ( ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
                    = ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
                 => ( ( D @ ( B @ ( E @ c_Divides_Odiv__class_Odiv ) ) )
                    = ( C @ ( A @ ( E @ c_Divides_Odiv__class_Odiv ) ) ) ) )
                & ( ( ( D @ ( B @ ( E @ c_Divides_Odiv__class_Odiv ) ) )
                    = ( C @ ( A @ ( E @ c_Divides_Odiv__class_Odiv ) ) ) )
                 => ( ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
                    = ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(976,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Omult__zero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Omult__zero) ).

thf(4616,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Omult__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(822,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__inverse2) ).

thf(4073,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(232,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_field__inverse) ).

thf(1801,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(530,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_even__less__0__iff) ).

thf(2856,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(1000,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring) ).

thf(4667,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(596,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__cancel) ).

thf(3138,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(159,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__lessD) ).

thf(1571,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(103,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zpower__zpower) ).

thf(1376,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(937,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oone,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oone) ).

thf(4509,plain,
    tc_Complex_Ocomplex @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(185,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__mono) ).

thf(1667,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(913,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__minus__eq__add) ).

thf(4430,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(472,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less) ).

thf(2632,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(948,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__smult__cancel) ).

thf(4541,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(465,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__divide__mult__cancel__left) ).

thf(2610,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(960,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__zero__iff) ).

thf(4569,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__vector )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Groups_Osgn__class_Osgn ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(319,axiom,
    tc_Int_Oint @ class_Rings_Oidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oidom) ).

thf(2111,plain,
    tc_Int_Oint @ class_Rings_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(919,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__eq__mod__eq__0) ).

thf(4452,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(610,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__assoc) ).

thf(3192,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(1027,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_psize__eq__0__iff) ).

thf(4726,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).

thf(554,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ominus__right) ).

thf(2974,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(19,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__le__power) ).

thf(1112,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(257,axiom,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__0__right) ).

thf(1877,plain,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(745,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__trans) ).

thf(3747,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(577,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__left__minus) ).

thf(3055,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Oab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(802,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__divide__mult__cancel__right) ).

thf(4009,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
          = ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(326,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__commute) ).

thf(2129,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(721,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oeq__refl) ).

thf(3668,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(819,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__commute) ).

thf(4064,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(283,axiom,
    tc_Int_Oint @ class_Rings_Oring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1) ).

thf(1993,plain,
    tc_Int_Oint @ class_Rings_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(512,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Power_Opower ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Power_Opower) ).

thf(2780,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(369,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__mono2) ).

thf(2267,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(807,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Osemiring )
     => ( ( A @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_combine__common__factor) ).

thf(4028,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Osemiring )
     => ( ( A @ ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(795,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
            = ( C @ c_Groups_Oone__class_Oone ) )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_right__inverse__eq) ).

thf(3977,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( A = B )
           => ( ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
              = ( C @ c_Groups_Oone__class_Oone ) ) )
          & ( ( ( B @ ( A @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
              = ( C @ c_Groups_Oone__class_Oone ) )
           => ( A = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(245,axiom,
    tc_Int_Oint @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ozero__neq__one) ).

thf(1838,plain,
    tc_Int_Oint @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(514,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__0__right) ).

thf(2785,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(564,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__one__right) ).

thf(3006,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Omonoid__mult )
     => ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(992,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__poly__code_I1_J) ).

thf(4652,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(534,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Osmult ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__1__left) ).

thf(2874,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Polynomial_Osmult ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(682,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Oadd__left) ).

thf(3483,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(794,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
        & ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
            = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zminus1__eq__if) ).

thf(3971,plain,
    ! [A: $i,B: $i] :
      ( ( B
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
           != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
        & ( ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
            = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
            = ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(91,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( B @ c_Nat_OSuc )
         != A )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__lessI) ).

thf(1344,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( B @ c_Nat_OSuc )
         != A )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(156,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__div__equality2) ).

thf(1556,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(454,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__eq__if) ).

thf(2542,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(394,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__le__nonneg__dividend) ).

thf(2329,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(1009,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( B @ ( C @ ( A @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__dvd) ).

thf(4684,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A
           != ( D @ c_Groups_Ozero__class_Ozero ) )
         => ( B @ ( C @ ( A @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).

thf(496,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Deriv_Oadd__diff__add) ).

thf(2710,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(954,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_OpCons ) ) )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pCons__0__0) ).

thf(4554,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Polynomial_OpCons ) ) )
        = ( A @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(681,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__add__eq__iff) ).

thf(3480,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(552,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__mod__minus__left) ).

thf(2962,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(402,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mod__equality2) ).

thf(2350,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(353,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__not__refl2) ).

thf(2212,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(149,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) ).

thf(1528,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(589,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( ~ ( ( A
                = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
              & ( B
                = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( C @ c_Groups_Oone__class_Oone ) ) )
        & ( ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__monic) ).

thf(3110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( ~ ( ( A
                = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
              & ( B
                = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( C @ c_Groups_Oone__class_Oone ) ) )
        & ( ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(780,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__commute) ).

thf(3917,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(379,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mod__imp__dvd) ).

thf(2288,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(789,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
          = ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) ) )
       => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
           => ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__dvd__antisym) ).

thf(3953,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
          = ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) ) )
       => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(360,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__increasing__iff) ).

thf(2244,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(883,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) ).

thf(4304,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(982,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Osmult ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__0__left) ).

thf(4626,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Osmult ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(820,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Limits_Ominus__diff__minus) ).

thf(4067,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(306,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib2) ).

thf(2069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(1065,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring) ).

thf(4844,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(1057,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring) ).

thf(4832,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(46,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zminus__zminus) ).

thf(1198,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(1024,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Odvd ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Odvd) ).

thf(4720,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Odvd ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).

thf(247,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_semiring__div__class_Omod__div__equality_H) ).

thf(1841,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(51,axiom,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__nat__zero__code) ).

thf(1213,plain,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(763,axiom,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__add) ).

thf(3838,plain,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(525,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) )
          = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( B @ ( C @ c_Polynomial_Odegree ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_synthetic__div__eq__0__iff) ).

thf(2835,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( ( B @ ( C @ c_Polynomial_Odegree ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( C @ c_Polynomial_Odegree ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(934,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__minus__right) ).

thf(4502,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(899,axiom,
    tc_Nat_Onat @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4373,plain,
    tc_Nat_Onat @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(176,axiom,
    tc_Int_Oint @ class_Groups_Oab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__group__add) ).

thf(1643,plain,
    tc_Int_Oint @ class_Groups_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(826,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__def) ).

thf(4085,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(408,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( C @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc__less__one) ).

thf(2369,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless ) ) )
         => ( C @ c_Groups_Oone__class_Oone @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(895,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ocomm__neutral) ).

thf(4362,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__mult )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(206,axiom,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__mult) ).

thf(1726,plain,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__add__self2) ).

thf(1364,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( C @ c_Groups_Oone__class_Oone @ ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(416,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__irrefl__nat) ).

thf(2390,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(558,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__replicate__append) ).

thf(2984,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(276,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(1973,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oab__semigroup__mult )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(446,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__inverse__eq__divide) ).

thf(2506,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(925,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__pCons) ).

thf(4473,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(565,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) )
        = ( A @ ( C @ c_Polynomial_Ocoeff ) @ ( B @ ( C @ c_Nat_Onat_Onat__case ) ) @ ( C @ c_Polynomial_OAbs__poly ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pCons__def) ).

thf(3009,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) )
        = ( A @ ( C @ c_Polynomial_Ocoeff ) @ ( B @ ( C @ c_Nat_Onat_Onat__case ) ) @ ( C @ c_Polynomial_OAbs__poly ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(270,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__antisym) ).

thf(1948,plain,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(673,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( D = C )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__eq__diff__eq) ).

thf(3444,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oab__group__add )
     => ( ( ( C @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) )
       => ( ( ( B = A )
           => ( D = C ) )
          & ( ( D = C )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(841,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ogroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ogroup__add) ).

thf(4136,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(872,axiom,
    tc_Nat_Onat @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Power_Opower) ).

thf(4260,plain,
    tc_Nat_Onat @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(636,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
      <=> ( ( B
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__two__disj) ).

thf(3293,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( ( B
              = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
            | ( B = A ) )
         => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
         => ( ( B
              = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
            | ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(89,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos) ).

thf(1340,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(894,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
          = ( B @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__equal__0__iff__equal) ).

thf(4355,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(1025,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__cancel__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring) ).

thf(4722,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__cancel__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).

thf(703,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__imp__neq) ).

thf(3571,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(288,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__add) ).

thf(2005,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(112,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__decreasing) ).

thf(1413,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( D @ c_Groups_Oone__class_Oone @ ( A @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(543,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__add__cancel) ).

thf(2914,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(521,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( A @ ( C @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_synthetic__div__correct) ).

thf(2816,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( A @ ( C @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Osynthetic__div ) ) @ ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(161,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__mult1) ).

thf(1583,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(920,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__smult) ).

thf(4458,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(481,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ominus__left) ).

thf(2665,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(1075,axiom,
    tc_Complex_Ocomplex @ class_Rings_Odivision__ring__inverse__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Odivision__ring__inverse__zero) ).

thf(4868,plain,
    tc_Complex_Ocomplex @ class_Rings_Odivision__ring__inverse__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(722,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__one__over) ).

thf(3672,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(115,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__lt__two__imp__zero__or__one) ).

thf(1420,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
        | ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(4,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__increasing) ).

thf(1081,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Oone__class_Oone @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(854,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__eq__0) ).

thf(4177,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(471,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__pCons) ).

thf(2629,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(282,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__self1__is__0) ).

thf(1990,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( C @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(349,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__zminus) ).

thf(2202,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(675,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__self__eq__0) ).

thf(3455,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(132,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Zero__not__Suc) ).

thf(1472,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(1011,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Groups_Ozero )
     => ( ( ( D @ ( F @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( F @ c_Groups_Ozero__class_Ozero @ ( E @ hAPP ) @ hAPP ) @ hAPP ) )
          = D )
       => ( ( A @ ( B @ ( F @ c_Polynomial_OpCons ) ) @ ( E @ ( D @ ( F @ ( C @ c_Polynomial_Opoly__rec ) ) ) ) )
          = ( A @ ( E @ ( D @ ( F @ ( C @ c_Polynomial_Opoly__rec ) ) ) ) @ ( A @ ( B @ ( E @ hAPP ) @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__rec__pCons) ).

thf(4688,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Groups_Ozero )
     => ( ( ( D @ ( F @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( F @ c_Groups_Ozero__class_Ozero @ ( E @ hAPP ) @ hAPP ) @ hAPP ) )
          = D )
       => ( ( A @ ( B @ ( F @ c_Polynomial_OpCons ) ) @ ( E @ ( D @ ( F @ ( C @ c_Polynomial_Opoly__rec ) ) ) ) )
          = ( A @ ( E @ ( D @ ( F @ ( C @ c_Polynomial_Opoly__rec ) ) ) ) @ ( A @ ( B @ ( E @ hAPP ) @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).

thf(834,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__0__right) ).

thf(4113,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(457,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( ( B @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
          = A )
      <=> ( ( ( B
              = ( D @ c_Groups_Ozero__class_Ozero ) )
           => ( A
              = ( D @ c_Groups_Ozero__class_Ozero ) ) )
          & ( ( B
             != ( D @ c_Groups_Ozero__class_Ozero ) )
           => ( C
              = ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__eq__eq) ).

thf(2564,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield__inverse__zero )
     => ( ( ( ( ( B
                = ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( A
                = ( D @ c_Groups_Ozero__class_Ozero ) ) )
            & ( ( B
               != ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( C
                = ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) )
         => ( ( B @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
            = A ) )
        & ( ( ( B @ ( C @ ( D @ c_Rings_Oinverse__class_Odivide ) ) )
            = A )
         => ( ( ( B
                = ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( A
                = ( D @ c_Groups_Ozero__class_Ozero ) ) )
            & ( ( B
               != ( D @ c_Groups_Ozero__class_Ozero ) )
             => ( C
                = ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(461,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__diff__eq__iff) ).

thf(2597,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(963,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(4581,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(1055,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult) ).

thf(4812,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).

thf(154,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zmult__distrib) ).

thf(1551,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(1032,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult) ).

thf(4749,plain,
    tc_Complex_Ocomplex @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).

thf(214,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__eq) ).

thf(1748,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(810,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__diff__right__eq) ).

thf(4035,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(765,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__eq__1__iff) ).

thf(3842,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(479,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__diff__nat) ).

thf(2661,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(345,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__right__mono) ).

thf(2192,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(617,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__minus) ).

thf(3214,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(166,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__self1__is__m) ).

thf(1605,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(502,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
    <=> ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oeq__iff) ).

thf(2732,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(593,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__mult__distrib) ).

thf(3129,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(340,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__pred_H) ).

thf(2175,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A
        = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(58,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__less__add1) ).

thf(1230,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(813,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__dvd__cancel) ).

thf(4045,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(995,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ozero) ).

thf(4658,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(231,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__inverse) ).

thf(1798,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(383,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__reduce) ).

thf(2298,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(741,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__pos__nat) ).

thf(3737,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(222,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__zless__mono2) ).

thf(1770,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(532,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).

thf(2863,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(640,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_times__divide__times__eq) ).

thf(3315,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Fields_Ofield__inverse__zero )
     => ( ( A @ ( B @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ ( E @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(555,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) ).

thf(2977,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(168,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonneg__nonneg) ).

thf(1611,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(56,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__mono) ).

thf(1225,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Oordered__semiring )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(875,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        | ( C = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__cancel2) ).

thf(4272,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( C = A ) )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(212,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__mono1) ).

thf(1743,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(226,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neq0__conv) ).

thf(1780,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( A
         != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(1035,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero )
          = A )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__reorient) ).

thf(4754,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ c_Groups_Ozero__class_Ozero )
            = A ) )
        & ( ( ( B @ c_Groups_Ozero__class_Ozero )
            = A )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).

thf(578,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Oprod__diff__prod) ).

thf(3058,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( D @ ( E @ c_Groups_Ominus__class_Ominus ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(415,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__iff1) ).

thf(2386,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(150,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ~ ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__sum__squares__lt__zero) ).

thf(1531,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ~ ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(888,axiom,
    ! [A: $i,B: $i] :
      ( ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
    <=> ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
        | ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__is__add) ).

thf(4324,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) )
       => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
      & ( ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
          | ( ( A
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(504,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__gr__0) ).

thf(2747,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          | ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(886,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_times__divide__eq__right) ).

thf(4314,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(957,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__poly__code_I2_J) ).

thf(4561,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(869,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oorder__refl) ).

thf(4240,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(492,axiom,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(2700,plain,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(952,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide_Ozero) ).

thf(4549,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__field )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(1038,axiom,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__mult) ).

thf(4768,plain,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).

thf(60,axiom,
    ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) )
    = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zminus__0) ).

thf(1235,plain,
    ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) )
    = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(373,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zdiv__equality_H) ).

thf(2276,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(893,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__double__add__iff__zero__le__single__add) ).

thf(4351,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(6,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__mono) ).

thf(1085,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(493,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__0) ).

thf(2701,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(26,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__neqE__nat) ).

thf(1130,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(909,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__div__neg) ).

thf(4409,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Oring__div )
     => ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = ( B @ ( A @ ( C @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(623,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A
          = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_equal__neg__zero) ).

thf(3246,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) )
        & ( ( A
            = ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(228,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__Suc__shift) ).

thf(1790,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(281,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zadd1__eq) ).

thf(1987,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(203,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__SucE) ).

thf(1716,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(786,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__minus) ).

thf(3940,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(858,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring) ).

thf(4190,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(338,axiom,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__1__right) ).

thf(2165,plain,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(236,axiom,
    ! [A: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_odd__less__0) ).

thf(1813,plain,
    ! [A: $i] :
      ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(79,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mult__cancel1) ).

thf(1300,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B = A )
       => ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
      & ( ( ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(174,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__inverse__inverse__eq) ).

thf(1637,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(488,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zminus2__not__zero) ).

thf(2688,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(614,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__not__sym) ).

thf(3202,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(864,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( B @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__times) ).

thf(4222,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Osgn__class_Osgn ) )
        = ( A @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( B @ ( C @ c_Groups_Osgn__class_Osgn ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(715,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( ( B
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) )
            = ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) )
        & ( ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) )
            = ( A @ ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Polynomial_Osmult ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__code) ).

thf(3623,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( ( B
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) )
            = ( B @ ( A @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) ) ) )
        & ( ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Polynomial_Opoly__gcd ) ) )
            = ( A @ ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Polynomial_Osmult ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(666,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__dvd__iff) ).

thf(3412,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( C @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(1042,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Omonom ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monom__eq__0) ).

thf(4781,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Omonom ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(297,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__assoc) ).

thf(2044,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(73,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__Suc2) ).

thf(1285,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(210,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ~ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__square__less__zero) ).

thf(1738,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__ring )
     => ~ ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(849,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__0__eq__0) ).

thf(4164,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(404,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pow__divides__eq__int) ).

thf(2354,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( C @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( C @ ( A @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(267,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__Suc__eq) ).

thf(1934,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(1051,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opcompose ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pcompose__0) ).

thf(4802,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ c_Polynomial_Opcompose ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).

thf(720,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ! [E: $i] :
                ( ( B @ ( E @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
               => ( ( A @ ( E @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
                 => ( C @ ( E @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) )
           => ( ( ( ~ ( ( A
                        = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
                      & ( B
                        = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
                 => ( ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
                    = ( D @ c_Groups_Oone__class_Oone ) ) )
                & ( ( ( A
                      = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
                    & ( B
                      = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
                 => ( ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
                    = ( D @ c_Groups_Ozero__class_Ozero ) ) ) )
             => ( ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) )
                = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__unique) ).

thf(3652,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ! [E: $i] :
                ( ( B @ ( E @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
               => ( ( A @ ( E @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
                 => ( C @ ( E @ ( D @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) )
           => ( ( ( ~ ( ( A
                        = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
                      & ( B
                        = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
                 => ( ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
                    = ( D @ c_Groups_Oone__class_Oone ) ) )
                & ( ( ( A
                      = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
                    & ( B
                      = ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
                 => ( ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
                    = ( D @ c_Groups_Ozero__class_Ozero ) ) ) )
             => ( ( A @ ( B @ ( D @ c_Polynomial_Opoly__gcd ) ) )
                = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(760,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Omonom ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_OpCons ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monom__0) ).

thf(3829,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Omonom ) ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_OpCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(840,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__pCons__Suc) ).

thf(4133,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ozero )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(915,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Ozero) ).

thf(4436,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(391,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zmult__distrib2) ).

thf(2320,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(142,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__dvd__cancel1) ).

thf(1504,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(792,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
      <=> ( C = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__right__cancel) ).

thf(3961,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__semigroup__add )
     => ( ( ( C = A )
         => ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
            = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) )
        & ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
            = ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(538,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__power__divide) ).

thf(2892,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(68,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__Suc) ).

thf(1272,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(527,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__cancel) ).

thf(2848,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(846,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( B @ ( C @ c_Polynomial_Ocoeff ) )
          = ( A @ ( C @ c_Polynomial_Ocoeff ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__inject) ).

thf(4150,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( B = A )
         => ( ( B @ ( C @ c_Polynomial_Ocoeff ) )
            = ( A @ ( C @ c_Polynomial_Ocoeff ) ) ) )
        & ( ( ( B @ ( C @ c_Polynomial_Ocoeff ) )
            = ( A @ ( C @ c_Polynomial_Ocoeff ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(483,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( H @ class_Fields_Ofield )
     => ( ( D @ ( E @ ( F @ ( G @ ( H @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( C @ ( E @ ( H @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( D @ ( A @ ( F @ ( H @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( H @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( F @ ( H @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( G @ ( H @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__mult) ).

thf(2675,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( H @ class_Fields_Ofield )
     => ( ( D @ ( E @ ( F @ ( G @ ( H @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( ( A @ ( B @ ( C @ ( E @ ( H @ c_Polynomial_Opdivmod__rel ) ) ) ) )
         => ( D @ ( A @ ( F @ ( H @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( H @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( F @ ( H @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( G @ ( H @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(301,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) ).

thf(2056,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(109,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zminus2) ).

thf(1404,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(424,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zmult2__eq) ).

thf(2411,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(685,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
    <=> ( C = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__right__cancel) ).

thf(3488,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C = A )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( C = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(198,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__1) ).

thf(1702,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(412,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__cancel) ).

thf(2379,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(54,axiom,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__semigroup__add) ).

thf(1221,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(930,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__eq__divide) ).

thf(4488,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(94,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__less__mono) ).

thf(1353,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(374,axiom,
    tc_Int_Oint @ class_Rings_Oring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring) ).

thf(2279,plain,
    tc_Int_Oint @ class_Rings_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(938,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__group__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__group__add) ).

thf(4510,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(273,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ? [C: $i] :
          ( A
          = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__iff__Suc__add) ).

thf(1955,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(333,axiom,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_lessI) ).

thf(2145,plain,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(531,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Divides_Oring__div )
     => ( ( ( D @ ( E @ ( F @ c_Divides_Odiv__class_Omod ) ) )
          = ( D @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( ( D @ ( B @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
         => ( ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( C @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__diff__cong) ).

thf(2860,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Divides_Oring__div )
     => ( ( ( D @ ( E @ ( F @ c_Divides_Odiv__class_Omod ) ) )
          = ( D @ ( C @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
       => ( ( ( D @ ( B @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( F @ c_Divides_Odiv__class_Omod ) ) ) )
         => ( ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) )
            = ( D @ ( A @ ( C @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Divides_Odiv__class_Omod ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(500,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( A @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__add__less) ).

thf(2722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocomm__monoid__add )
     => ( ( B @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( A @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(787,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_right__minus__eq) ).

thf(3943,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( B = A )
         => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(865,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( C @ ( D @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( B @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__dvd__mono) ).

thf(4225,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( C @ ( D @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( B @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(458,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divisors__zero) ).

thf(2584,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ono__zero__divisors )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(1003,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( F @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ ( F @ ( E @ c_Polynomial_Opoly__rec ) ) ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( E @ c_Polynomial_Opoly__rec ) ) ) ) @ ( D @ ( F @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_fequal ) @ ( E @ c_If ) ) ) @ ( A @ ( B @ ( C @ hAPP ) @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__rec_Osimps) ).

thf(4671,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( F @ c_Polynomial_OpCons ) ) @ ( C @ ( D @ ( F @ ( E @ c_Polynomial_Opoly__rec ) ) ) ) )
        = ( A @ ( C @ ( D @ ( F @ ( E @ c_Polynomial_Opoly__rec ) ) ) ) @ ( D @ ( F @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ c_fequal ) @ ( E @ c_If ) ) ) @ ( A @ ( B @ ( C @ hAPP ) @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(634,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B != A )
       => ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Ole__neq__trans) ).

thf(3284,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B != A )
       => ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(661,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ ( E @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( B @ ( A @ ( F @ c_Polynomial_Osmult ) ) @ ( C @ ( A @ ( F @ c_Polynomial_Osmult ) ) @ ( D @ ( E @ ( A @ ( F @ c_Polynomial_Osmult ) ) @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__smult__left) ).

thf(3400,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Fields_Ofield )
     => ( ( B @ ( C @ ( D @ ( E @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) )
       => ( B @ ( A @ ( F @ c_Polynomial_Osmult ) ) @ ( C @ ( A @ ( F @ c_Polynomial_Osmult ) ) @ ( D @ ( E @ ( A @ ( F @ c_Polynomial_Osmult ) ) @ ( F @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(878,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__asym_H) ).

thf(4289,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ~ ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(246,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sum__squares__ge__zero) ).

thf(1839,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__ring )
     => ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(674,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) ).

thf(3452,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__ring__1 )
     => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
      <=> ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__iff__poly__eq__0) ).

thf(3565,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) )
        & ( ( A @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(128,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
      | ~ ( A @ ( B @ c_fequal ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__fequal__1) ).

thf(1459,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
      | ~ ( A @ ( B @ c_fequal ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(696,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( C @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel) ).

thf(3533,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( C @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(248,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__iff2) ).

thf(1844,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oordered__ring )
     => ( ( ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( D @ ( E @ ( B @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(346,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zminus__zminus) ).

thf(2194,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(956,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring) ).

thf(4559,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(52,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Odivision__ring )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__1) ).

thf(1216,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Odivision__ring )
     => ( ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(24,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semidom ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semidom) ).

thf(1126,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(830,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__self3) ).

thf(4102,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(385,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult2__eq) ).

thf(2305,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
      = ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(478,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__1) ).

thf(2658,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(285,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__mono1) ).

thf(1998,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(470,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__poly__add__left) ).

thf(2626,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(842,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ~ ( B @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ c_Nat_OSuc @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__2) ).

thf(4138,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ~ ( B @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ c_Nat_OSuc @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(1040,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) )
          = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__zero__iff) ).

thf(4770,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( ( ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
            & ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(275,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel1) ).

thf(1966,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( B = A )
         => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(33,axiom,
    tc_Int_Oint @ class_Rings_Oordered__cancel__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__cancel__semiring) ).

thf(1157,plain,
    tc_Int_Oint @ class_Rings_Oordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(911,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ozero__right) ).

thf(4419,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_RealVector_Oreal__normed__algebra )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(243,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
      | ( B = A )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zless__linear) ).

thf(1834,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
      | ( B = A )
      | ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(994,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add) ).

thf(4656,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(401,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power_Opower_Opower__Suc) ).

thf(2347,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( C @ ( D @ ( E @ c_Power_Opower_Opower ) ) @ hAPP ) @ hAPP ) @ ( B @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(181,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__inverse) ).

thf(1654,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(75,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__add__self1) ).

thf(1289,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( A @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(126,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__commutes) ).

thf(1453,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Omonoid__mult )
     => ( ( B @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(208,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__less__imp__less) ).

thf(1734,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( B @ ( A @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(885,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( E
         != ( F @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( B != A )
            & ( D = C ) )
         => ( ( B @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
           != ( A @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__scale__eq__noteq) ).

thf(4310,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( E
         != ( F @ c_Groups_Ozero__class_Ozero ) )
       => ( ( ( B != A )
            & ( D = C ) )
         => ( ( B @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
           != ( A @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(234,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__minus__eq) ).

thf(1807,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(818,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__inverse__minus__eq) ).

thf(4061,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(70,axiom,
    ! [A: $i,B: $i] :
      ( B
      = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zdiv__equality) ).

thf(1276,plain,
    ! [A: $i,B: $i] :
      ( B
      = ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(86,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
            = ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__inverse__mult__distrib) ).

thf(1326,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
            = ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(460,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__mult__distrib) ).

thf(2594,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(133,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__imp__zdiv__neg__iff) ).

thf(1476,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(189,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mod__equality) ).

thf(1674,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( B @ ( C @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(456,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__no__zero__divisors )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__0__iff) ).

thf(2554,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring__no__zero__divisors )
     => ( ( ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A
              = ( C @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(644,axiom,
    ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_One__nat__def) ).

thf(3336,plain,
    ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(839,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( B
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( A @ ( C @ c_Polynomial_Odegree ) ) ) )
        & ( ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__smult__eq) ).

thf(4126,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( B
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( A @ ( C @ c_Polynomial_Odegree ) ) ) )
        & ( ( B
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ c_Polynomial_Odegree ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(213,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__monom) ).

thf(1745,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Omonom ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(716,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ c_Polynomial_Opos__poly ) )
      <=> ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__poly__def) ).

thf(3630,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ c_Polynomial_Opos__poly ) ) )
        & ( ( A @ ( B @ c_Polynomial_Opos__poly ) )
         => ( A @ ( B @ c_Polynomial_Odegree ) @ ( A @ ( B @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(251,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__less__power) ).

thf(1854,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(367,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__mod__bound) ).

thf(2262,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(714,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
    <=> ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        | ( C
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__dvd__cancel__disj) ).

thf(3613,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          | ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          | ( C
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(495,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__add__right) ).

thf(2707,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(380,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__by__0) ).

thf(2290,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(362,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_right__inverse) ).

thf(2251,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A
         != ( B @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( B @ c_Groups_Oone__class_Oone ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(999,axiom,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring__1) ).

thf(4666,plain,
    tc_Complex_Ocomplex @ class_Rings_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(328,axiom,
    tc_Int_Oint @ class_Divides_Oring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Divides_Oring__div) ).

thf(2135,plain,
    tc_Int_Oint @ class_Divides_Oring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(1007,axiom,
    tc_Complex_Ocomplex @ class_Groups_Ogroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ogroup__add) ).

thf(4681,plain,
    tc_Complex_Ocomplex @ class_Groups_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(143,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__left__eq) ).

thf(1508,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(1021,axiom,
    tc_Complex_Ocomplex @ class_Rings_Odivision__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Odivision__ring) ).

thf(4715,plain,
    tc_Complex_Ocomplex @ class_Rings_Odivision__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).

thf(588,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__eq__0__iff__dvd) ).

thf(3104,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(287,axiom,
    tc_Int_Oint @ class_Groups_Oone,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oone) ).

thf(2004,plain,
    tc_Int_Oint @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(563,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) ).

thf(3003,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(507,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__imp__not__eq) ).

thf(2764,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(238,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__power) ).

thf(1822,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( A @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(1058,axiom,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__add) ).

thf(4834,plain,
    tc_Complex_Ocomplex @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(693,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__eq__diff__pred) ).

thf(3516,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(1018,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
          = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      <=> ( B
          = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_offset__poly__eq__0__iff) ).

thf(4705,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).

thf(768,axiom,
    tc_Nat_Onat @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ozero__neq__one) ).

thf(3859,plain,
    tc_Nat_Onat @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(873,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
         != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
      & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__zminus2__eq__if) ).

thf(4261,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
         != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ) )
      & ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(581,axiom,
    tc_Nat_Onat @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Omult__zero) ).

thf(3085,plain,
    tc_Nat_Onat @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(138,axiom,
    tc_Int_Oint @ class_Rings_Oordered__comm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__comm__semiring) ).

thf(1491,plain,
    tc_Int_Oint @ class_Rings_Oordered__comm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(438,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( ( ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) )
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          | ( B
            = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__root) ).

thf(2475,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( ( ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) )
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Polynomial_Opoly ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) )
             != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
            | ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(860,axiom,
    tc_Nat_Onat @ class_Rings_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring) ).

thf(4197,plain,
    tc_Nat_Onat @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(629,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__minus__self__iff) ).

thf(3267,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(122,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( A @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
           => ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__antisym__nonneg) ).

thf(1440,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( A @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(80,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult1__eq) ).

thf(1308,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(485,axiom,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gcd__lcm__complete__lattice__nat_Obot__least) ).

thf(2680,plain,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(798,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__0__left) ).

thf(3993,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(540,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__add__left) ).

thf(2902,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(719,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) )
      <=> ( A
          = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__eq__0__iff) ).

thf(3645,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( ( A
            = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
         => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) )
        & ( ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( A
            = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(978,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add) ).

thf(4619,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocancel__comm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(35,axiom,
    ! [A: $i] :
      ( A
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_n__not__Suc__n) ).

thf(1161,plain,
    ! [A: $i] :
      ( A
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(1030,axiom,
    tc_Complex_Ocomplex @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Omult__zero) ).

thf(4745,plain,
    tc_Complex_Ocomplex @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(433,axiom,
    ! [A: $i,B: $i] :
      ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__le) ).

thf(2447,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) )
      & ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(477,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
        & ( B
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__is__0) ).

thf(2648,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
          & ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(707,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( B @ ( C @ ( E @ c_Polynomial_Osmult ) ) @ ( D @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ ( A @ ( E @ c_Polynomial_OpCons ) ) ) )
       => ( ( B
            = ( C @ ( D @ ( E @ c_Polynomial_Osynthetic__div ) ) ) )
          & ( A
            = ( C @ ( D @ ( E @ c_Polynomial_Opoly ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_synthetic__div__unique) ).

thf(3585,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__0 )
     => ( ( ( B @ ( C @ ( E @ c_Polynomial_Osmult ) ) @ ( D @ ( E @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
          = ( B @ ( A @ ( E @ c_Polynomial_OpCons ) ) ) )
       => ( ( B
            = ( C @ ( D @ ( E @ c_Polynomial_Osynthetic__div ) ) ) )
          & ( A
            = ( C @ ( D @ ( E @ c_Polynomial_Opoly ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(225,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__mult1) ).

thf(1777,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( C
         != ( D @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Divides_Odiv__class_Odiv ) ) )
          = ( A @ ( B @ ( D @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(903,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__distrib) ).

thf(4387,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(305,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__mult__self1) ).

thf(2066,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( C @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(1,conjecture,
    ( v_q
    = ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).

thf(2,negated_conjecture,
    ( v_q
   != ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(1077,plain,
    ( v_q
   != ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(406,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__not__refl3) ).

thf(2362,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(337,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__add__right__eq) ).

thf(2162,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Divides_Osemiring__div )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( A @ ( B @ ( D @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(1076,axiom,
    tc_Complex_Ocomplex @ class_Rings_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring) ).

thf(4869,plain,
    tc_Complex_Ocomplex @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(544,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ozero )
     => ( ( ( C @ ( D @ ( E @ c_Polynomial_OpCons ) ) )
          = ( A @ ( B @ ( E @ c_Polynomial_OpCons ) ) ) )
      <=> ( ( C = A )
          & ( D = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pCons__eq__iff) ).

thf(2917,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ozero )
     => ( ( ( ( C = A )
            & ( D = B ) )
         => ( ( C @ ( D @ ( E @ c_Polynomial_OpCons ) ) )
            = ( A @ ( B @ ( E @ c_Polynomial_OpCons ) ) ) ) )
        & ( ( ( C @ ( D @ ( E @ c_Polynomial_OpCons ) ) )
            = ( A @ ( B @ ( E @ c_Polynomial_OpCons ) ) ) )
         => ( ( C = A )
            & ( D = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(284,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__n__not__n) ).

thf(1994,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(700,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_left__add__mult__distrib) ).

thf(3552,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
      = ( A @ ( C @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(410,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos2) ).

thf(2374,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__cancel__semiring )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(1049,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__add) ).

thf(4794,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(172,axiom,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Osemiring__div ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Divides_Osemiring__div) ).

thf(1625,plain,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(628,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_RealVector_Oreal__field )
     => ( ( A @ ( B @ ( C @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ ( A @ ( C @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Rings_Oinverse__class_Odivide ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_DERIV__mult__lemma) ).

thf(3264,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_RealVector_Oreal__field )
     => ( ( A @ ( B @ ( C @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ ( A @ ( C @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Rings_Oinverse__class_Odivide ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( D @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Rings_Oinverse__class_Odivide ) ) @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(450,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( A @ ( A @ ( B @ ( C @ c_Polynomial_Oorder ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__1) ).

thf(2522,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( A @ ( A @ ( B @ ( C @ c_Polynomial_Oorder ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(1064,axiom,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1) ).

thf(4843,plain,
    tc_Complex_Ocomplex @ class_RealVector_Oreal__normed__algebra__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).

thf(188,axiom,
    c_fTrue @ hBOOL,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__fTrue__1) ).

thf(1673,plain,
    c_fTrue @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(594,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__pCons__left) ).

thf(3132,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(332,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gr0I) ).

thf(2142,plain,
    ! [A: $i] :
      ( ( A
       != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(656,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__diff__left) ).

thf(3379,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(838,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring) ).

thf(4125,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(365,axiom,
    tc_Int_Oint @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__semigroup__add) ).

thf(2258,plain,
    tc_Int_Oint @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(27,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__by__1) ).

thf(1133,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Divides_Odiv__class_Odiv ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(924,axiom,
    tc_Nat_Onat @ class_Groups_Oordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(4472,plain,
    tc_Nat_Onat @ class_Groups_Oordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(907,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Oadd) ).

thf(4399,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(1010,axiom,
    tc_Complex_Ocomplex @ class_Int_Oring__char__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Int_Oring__char__0) ).

thf(4687,plain,
    tc_Complex_Ocomplex @ class_Int_Oring__char__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).

thf(769,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__1) ).

thf(3860,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(691,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__add) ).

thf(3511,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(1002,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring) ).

thf(4670,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(191,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__zminus2) ).

thf(1678,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ouminus__class_Ouminus ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(557,axiom,
    tc_Nat_Onat @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__mult) ).

thf(2983,plain,
    tc_Nat_Onat @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(499,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) ).

thf(2719,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
        = ( B @ ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(12,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__less__power__Suc) ).

thf(1097,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(459,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( ( B
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Oone__class_Oone ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__num__eq__if) ).

thf(2587,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Power_Opower )
     => ( ( ( B
           != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( B
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
         => ( ( B @ ( A @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
            = ( C @ c_Groups_Oone__class_Oone ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(463,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ c_Polynomial_Odegree ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__monom__eq) ).

thf(2604,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ c_Polynomial_Odegree ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(677,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__right__mono) ).

thf(3464,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(105,axiom,
    ! [A: $i] :
      ( ( A
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__self) ).

thf(1380,plain,
    ! [A: $i] :
      ( ( A
       != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
     => ( ( A @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) )
        = ( tc_Int_Oint @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(289,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__add__self2) ).

thf(2008,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(1062,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oring__no__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors) ).

thf(4841,plain,
    tc_Complex_Ocomplex @ class_Rings_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(686,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__left) ).

thf(3496,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oring )
     => ( ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( A @ ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(931,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A
           != ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( A @ ( B @ c_Polynomial_Odegree ) @ c_Nat_OSuc ) ) )
        & ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_psize__def) ).

thf(4491,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ozero )
     => ( ( ( A
           != ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( A @ ( B @ c_Polynomial_Odegree ) @ c_Nat_OSuc ) ) )
        & ( ( A
            = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Fundamental__Theorem__Algebra__Mirabelle_Opsize ) )
            = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(217,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonpos__nonpos) ).

thf(1757,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oordered__ring )
     => ( ( C @ c_Groups_Ozero__class_Ozero @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ c_Groups_Ozero__class_Ozero @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(962,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__mult) ).

thf(4579,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oab__semigroup__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

thf(508,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__smult__left) ).

thf(2768,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Odiv ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(728,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
          = ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
      <=> ( ( B
            = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_square__eq__iff) ).

thf(3690,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( ( ( B
              = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
            | ( B = A ) )
         => ( ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) )
        & ( ( ( B @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( A @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
         => ( ( B
              = ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) ) )
            | ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(134,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__self) ).

thf(1480,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(567,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( A @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__gcd__dvd2) ).

thf(3015,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Ofield )
     => ( A @ ( A @ ( B @ ( C @ c_Polynomial_Opoly__gcd ) ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(790,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__diff__left) ).

thf(3956,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Osmult ) ) @ ( A @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(642,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult2) ).

thf(3328,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(892,axiom,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) ).

thf(4350,plain,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(494,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).

thf(2704,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(111,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) ).

thf(1410,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(407,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__0) ).

thf(2366,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(201,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).

thf(1710,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(422,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__cancel__right) ).

thf(2404,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(262,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_division__ring__inverse__add) ).

thf(1891,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A
           != ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(468,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__inverse) ).

thf(2619,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(639,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ~ ( B @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ c_Nat_OSuc @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order) ).

thf(3309,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ~ ( B @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ c_Nat_OSuc @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
          & ( B @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( C @ c_Groups_Oone__class_Oone @ ( C @ c_Polynomial_OpCons ) ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(759,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( B = A )
      <=> ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__iff__diff__eq__0) ).

thf(3822,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Oab__group__add )
     => ( ( ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( A @ ( B @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(182,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__mult__div__cancel) ).

thf(1657,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(396,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__imp__le__right) ).

thf(2334,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Oordered__ab__semigroup__add__imp__le )
     => ( ( B @ ( A @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(647,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__less__divisor) ).

thf(3347,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(877,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__ab__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__imp__eq) ).

thf(4286,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ocancel__ab__semigroup__add )
     => ( ( ( B @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( C @ ( D @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(847,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( C @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oless__le__trans) ).

thf(4157,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( C @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(194,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__zdiffD) ).

thf(1693,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
       => ( B @ ( C @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(1073,axiom,
    ( v_p
    = ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pe) ).

thf(4865,plain,
    ( v_p
    = ( tc_Complex_Ocomplex @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(917,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
          = A )
      <=> ( A
          = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__equal__zero) ).

thf(4442,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Olinordered__ab__group__add )
     => ( ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
            = A ) )
        & ( ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
            = A )
         => ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(750,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diffs0__imp__equal) ).

thf(3765,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
     => ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(482,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( A
           != ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) )
        & ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__self__if) ).

thf(2668,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring__inverse__zero )
     => ( ( ( A
           != ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
            = ( B @ c_Groups_Oone__class_Oone ) ) )
        & ( ( A
            = ( B @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( A @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
            = ( B @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(36,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Divides_Otransfer__nat__int__function__closures_I1_J) ).

thf(1165,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(294,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ hAPP ) @ hAPP ) @ hBOOL )
      <=> ! [D: $i,E: $i] :
            ( ( ( A
                = ( E @ ( D @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
              & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
              & ( E @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
           => ( E @ ( D @ ( B @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_split__neg__lemma) ).

thf(2021,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( ! [D: $i,E: $i] :
              ( ( ( A
                  = ( E @ ( D @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
                & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
                & ( E @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
             => ( E @ ( D @ ( B @ hAPP ) @ hAPP ) @ hBOOL ) )
         => ( C @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ hAPP ) @ hAPP ) @ hBOOL ) )
        & ( ( C @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( A @ ( tc_Int_Oint @ c_Divides_Odiv__class_Odiv ) ) @ ( B @ hAPP ) @ hAPP ) @ hBOOL )
         => ! [D: $i,E: $i] :
              ( ( ( A
                  = ( E @ ( D @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
                & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
                & ( E @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) )
             => ( E @ ( D @ ( B @ hAPP ) @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(304,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__antisym) ).

thf(2063,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(336,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__assoc) ).

thf(2159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(241,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Divides_Osemiring__div )
     => ( ( C @ ( D @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( B @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( A @ ( E @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ ( C @ ( E @ c_Divides_Odiv__class_Odiv ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__div__if__dvd) ).

thf(1829,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Divides_Osemiring__div )
     => ( ( C @ ( D @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( B @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
         => ( ( B @ ( A @ ( E @ c_Divides_Odiv__class_Odiv ) ) @ ( D @ ( C @ ( E @ c_Divides_Odiv__class_Odiv ) ) @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
            = ( B @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Divides_Odiv__class_Odiv ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(571,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) )
      <=> ( ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__add__iff1) ).

thf(3025,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Oring )
     => ( ( ( ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
            = A )
         => ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) ) )
        & ( ( ( C @ ( D @ ( E @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
            = ( A @ ( D @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( ( C @ ( D @ ( B @ ( E @ ( F @ c_Groups_Ominus__class_Ominus ) ) @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(66,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__le__mono) ).

thf(1268,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Oordered__cancel__ab__semigroup__add )
     => ( ( C @ ( D @ ( E @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( D @ ( E @ c_Groups_Oplus__class_Oplus ) ) @ ( E @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(870,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
          = ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) )
      <=> ( ( C = A )
          | ( D = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_crossproduct__eq) ).

thf(4242,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( ( C = A )
            | ( D = B ) )
         => ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) ) )
        & ( ( ( A @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) )
            = ( C @ ( B @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( D @ ( E @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( ( C = A )
            | ( D = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(772,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__cancel) ).

thf(3869,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B = A )
       => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) )
      & ( ( ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
          = ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(277,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__self1__is__id) ).

thf(1976,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( A @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(97,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) ).

thf(1357,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(569,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__right) ).

thf(3020,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(698,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( B
          = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) )
      <=> ( A
          = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0__iff) ).

thf(3538,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct )
     => ( ( ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( B
            = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) )
        & ( ( B
            = ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( A
            = ( C @ c_Groups_Ozero__class_Ozero ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(777,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__minus) ).

thf(3886,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Groups_Ouminus__class_Ouminus ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(258,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__0__less__mult__iff) ).

thf(1880,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(659,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__smult__left) ).

thf(3394,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
        = ( A @ ( B @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(827,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__add1__eq) ).

thf(4088,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
      = ( A @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(10,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__one) ).

thf(1093,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__semidom )
     => ( A @ c_Groups_Oone__class_Oone @ ( A @ c_Groups_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(403,axiom,
    tc_Int_Oint @ class_Groups_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) ).

thf(2353,plain,
    tc_Int_Oint @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(444,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__mult__degree__sum) ).

thf(2502,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
        = ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(359,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__div__cancel1) ).

thf(2241,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) )
        = ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Odiv ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(325,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__less__imp__less) ).

thf(2127,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semiring )
     => ( ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(462,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odvd )
     => ( ( C
          = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( C @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvdI) ).

thf(2600,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Odvd )
     => ( ( C
          = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) )
       => ( C @ ( B @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(901,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__mult__left) ).

thf(4382,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(440,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_field__class_Onormalizing__field__rules_I2_J) ).

thf(2485,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) )
        = ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Oinverse__class_Odivide ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(152,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nonzero__power__inverse) ).

thf(1537,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Odivision__ring )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( A @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Rings_Oinverse__class_Oinverse ) )
          = ( A @ ( B @ ( C @ c_Rings_Oinverse__class_Oinverse ) @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(453,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__add) ).

thf(2539,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ouminus__class_Ouminus ) )
        = ( B @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( A @ ( C @ c_Groups_Ouminus__class_Ouminus ) @ ( C @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(202,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
        = ( C @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__div__trivial) ).

thf(1713,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( A @ ( A @ ( B @ ( C @ c_Divides_Odiv__class_Omod ) ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
        = ( C @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(439,axiom,
    tc_Nat_Onat @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__semigroup__add) ).

thf(2484,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(1012,axiom,
    ( ( c_fFalse @ c_HOL_Obool_Obool__size )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_bool_Osize_I2_J) ).

thf(4691,plain,
    ( ( c_fFalse @ c_HOL_Obool_Obool__size )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).

thf(757,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( B = A )
      <=> ! [D: $i] :
            ( ( D @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( D @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_expand__poly__eq) ).

thf(3812,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ! [D: $i] :
              ( ( D @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
              = ( D @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) ) )
         => ( B = A ) )
        & ( ( B = A )
         => ! [D: $i] :
              ( ( D @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
              = ( D @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(646,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D @ class_Rings_Odvd )
        & ( D @ class_Rings_Ocomm__ring ) )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ! [E: $i,F: $i] :
            ( ( A @ ( E @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
          <=> ( A @ ( B @ ( F @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inf__period_I4_J) ).

thf(3341,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D @ class_Rings_Odvd )
        & ( D @ class_Rings_Ocomm__ring ) )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ! [E: $i,F: $i] :
            ( ( ( A @ ( B @ ( F @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
             => ( A @ ( E @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) )
            & ( ( A @ ( E @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
             => ( A @ ( B @ ( F @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(1004,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring__0) ).

thf(4674,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(965,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__0) ).

thf(4584,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(1043,axiom,
    tc_Complex_Ocomplex @ class_Rings_Oidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oidom) ).

thf(4784,plain,
    tc_Complex_Ocomplex @ class_Rings_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).

thf(951,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__ring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__ring) ).

thf(4547,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(137,axiom,
    tc_Int_Oint @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__mult) ).

thf(1490,plain,
    tc_Int_Oint @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(310,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib) ).

thf(2087,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(1070,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Polynomial_Opos__poly ) )
        | ( A @ ( B @ c_Polynomial_Opos__poly ) )
        | ( A
          = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__poly__total) ).

thf(4858,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Olinordered__idom )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ouminus__class_Ouminus ) @ ( B @ c_Polynomial_Opos__poly ) )
        | ( A @ ( B @ c_Polynomial_Opos__poly ) )
        | ( A
          = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).

thf(993,axiom,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__mult) ).

thf(4655,plain,
    tc_Complex_Ocomplex @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(474,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Odvd__class_Odvd ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__dvd) ).

thf(2640,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Rings_Odvd__class_Odvd ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(788,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = ( C @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( A @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( A @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__pCons) ).

thf(3950,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Fields_Ofield )
     => ( ( C
         != ( D @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( ( C @ ( A @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) )
          = ( C @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( C @ ( D @ c_Polynomial_Odegree ) @ ( C @ ( A @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ c_Polynomial_Ocoeff ) @ hAPP ) @ ( D @ c_Rings_Oinverse__class_Odivide ) ) @ ( D @ c_Polynomial_Osmult ) ) @ ( C @ ( A @ ( D @ tc_Polynomial_Opoly @ c_Divides_Odiv__class_Omod ) ) @ ( B @ ( D @ c_Polynomial_OpCons ) ) @ ( D @ tc_Polynomial_Opoly @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(1041,axiom,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_Onat_Onat__size )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat_Osize_I1_J) ).

thf(4779,plain,
    ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_Onat_Onat__size )
    = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).

thf(619,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Ole__less__trans) ).

thf(3227,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( ~ ( B @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(101,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmod__simps_I3_J) ).

thf(1367,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
      = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(268,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__comm__semiring__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) ).

thf(1944,plain,
    tc_Int_Oint @ class_Rings_Olinordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(1017,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ ( tc_Complex_Ocomplex @ ( tc_Complex_Ocomplex @ c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant ) ) )
     => ? [B: $i] :
          ( ( B @ ( A @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
          = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_fundamental__theorem__of__algebra) ).

thf(4702,plain,
    ! [A: $i] :
      ( ~ ( A @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ ( tc_Complex_Ocomplex @ ( tc_Complex_Ocomplex @ c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant ) ) )
     => ? [B: $i] :
          ( ( B @ ( A @ ( tc_Complex_Ocomplex @ c_Polynomial_Opoly ) @ hAPP ) )
          = ( tc_Complex_Ocomplex @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(269,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( C
         != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__mult__cancel) ).

thf(1945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( C
         != ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(616,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__add) ).

thf(3212,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(192,axiom,
    tc_Int_Oint @ class_Rings_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring) ).

thf(1681,plain,
    tc_Int_Oint @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(663,axiom,
    tc_Nat_Onat @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__add) ).

thf(3405,plain,
    tc_Nat_Onat @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(293,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_div__mult__self2__is__id) ).

thf(2018,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Divides_Osemiring__div )
     => ( ( B
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( ( B @ ( B @ ( A @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Divides_Odiv__class_Odiv ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(533,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( C @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( ( B = A )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd_Oord__less__eq__trans) ).

thf(2866,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ~ ( C @ ( B @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
        & ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) )
     => ( ( B = A )
       => ( ~ ( C @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
          & ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(848,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__diff__cancel) ).

thf(4161,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ogroup__add )
     => ( ( A @ ( A @ ( B @ ( C @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(524,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__poly__0__left) ).

thf(2832,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).

thf(1848,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(302,axiom,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__less0) ).

thf(2059,plain,
    ! [A: $i] :
      ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(576,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divide__zero__left) ).

thf(3052,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Odivision__ring )
     => ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Rings_Oinverse__class_Odivide ) ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(1072,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pdivmod__rel__by__0) ).

thf(4863,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Ofield )
     => ( A @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( B @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Polynomial_Opdivmod__rel ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(561,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__smult) ).

thf(2993,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( C @ ( D @ c_Polynomial_Osmult ) ) @ ( D @ c_Polynomial_Opoly ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Polynomial_Opoly ) @ hAPP ) @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(124,axiom,
    tc_Int_Oint @ class_Groups_Olinordered__ab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Olinordered__ab__group__add) ).

thf(1449,plain,
    tc_Int_Oint @ class_Groups_Olinordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(737,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__diff) ).

thf(3721,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__ring__1 )
     => ( ( B @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) )
         => ( A @ ( B @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( D @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(853,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Ocoeff ) )
        = ( A @ ( C @ c_Polynomial_Ocoeff ) @ ( B @ ( C @ c_Nat_Onat_Onat__case ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_coeff__pCons) ).

thf(4174,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Ocoeff ) )
        = ( A @ ( C @ c_Polynomial_Ocoeff ) @ ( B @ ( C @ c_Nat_Onat_Onat__case ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(556,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Omonom ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Omonom ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_smult__monom) ).

thf(2980,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__0 )
     => ( ( A @ ( B @ ( D @ c_Polynomial_Omonom ) ) @ ( C @ ( D @ c_Polynomial_Osmult ) ) )
        = ( A @ ( B @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Polynomial_Omonom ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(573,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__size) ).

thf(3035,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Nat_Onat @ c_Nat_Osize__class_Osize ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(704,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) ).

thf(3575,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Oone__class_Oone ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(953,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__1) ).

thf(4552,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(801,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Oidom )
        & ( C @ class_Int_Oring__char__0 ) )
     => ( ( ( B @ ( C @ c_Polynomial_Opoly ) )
          = ( A @ ( C @ c_Polynomial_Opoly ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_poly__eq__iff) ).

thf(4002,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C @ class_Rings_Oidom )
        & ( C @ class_Int_Oring__char__0 ) )
     => ( ( ( B = A )
         => ( ( B @ ( C @ c_Polynomial_Opoly ) )
            = ( A @ ( C @ c_Polynomial_Opoly ) ) ) )
        & ( ( ( B @ ( C @ c_Polynomial_Opoly ) )
            = ( A @ ( C @ c_Polynomial_Opoly ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(114,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__Suc) ).

thf(1417,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
      = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(254,axiom,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(1870,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(509,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Odiff__left) ).

thf(2771,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_RealVector_Oreal__normed__algebra )
     => ( ( A @ ( B @ ( C @ ( D @ c_Groups_Ominus__class_Ominus ) ) @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( A @ ( B @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( D @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(4971,plain,
    $false,
    inference(e,[status(thm)],[2094,3883,2072,3434,1237,3477,4015,3927,1586,1501,4201,2744,4835,2889,1190,4464,2341,2336,2031,3930,4169,1559,1105,2463,3397,2787,2499,1315,3863,1083,1982,1179,2331,4682,2280,1554,1110,3172,2077,3681,1596,4500,2622,2381,1443,2014,2114,4623,4592,4714,1158,2512,1522,3762,4515,2395,2908,2062,3377,2527,2248,4807,4697,3959,4603,3012,1205,1793,3402,2854,1533,3382,4479,1173,4105,1608,3745,2607,3135,4026,2316,3338,1275,4856,4564,3253,1640,1954,4762,1705,3017,1788,1426,3217,3530,4839,3367,4110,4528,3175,4650,2168,4058,1342,4294,1222,3717,2360,1095,4047,2216,3545,3634,3502,4679,4618,1613,1645,4198,4607,2400,3597,3688,1887,4371,4079,3281,3466,4665,2819,3880,2136,1518,4610,4637,3204,4524,3195,3275,4070,2269,2755,1871,3061,3148,1279,1832,1462,3713,4439,1287,1985,4422,1660,1298,2339,3046,2444,2217,1511,2274,2682,3125,1826,3749,2488,2420,2691,3099,4788,1147,1890,1347,4317,3835,1773,1494,4390,3231,3866,4669,2713,4119,1379,3374,3385,2403,3638,1087,1247,2049,1115,2929,3470,1931,1568,3734,4368,4141,1362,1963,4793,4507,1232,3258,2829,3788,2178,3431,4583,2996,2782,3163,4724,1677,2777,4810,4696,2237,1415,2616,4396,1664,2794,2105,1450,4018,2090,1109,4297,4733,2259,2880,1682,1136,4847,4147,4050,2122,1383,3357,2254,3923,3290,3986,4230,4842,2190,3505,3361,1433,2103,2965,1697,2154,4082,4718,4751,1169,4660,2437,1928,3189,3739,2281,1622,2371,1394,3157,1526,4307,2137,2286,4038,4643,1201,3406,2205,3582,4604,3591,3303,3720,3852,3186,4557,4076,4700,1651,3151,2011,3097,4589,4629,3984,3169,2047,3318,1708,2093,1311,2296,4576,4031,1736,2496,2790,1768,2905,4172,2000,2264,1091,2805,3603,1519,1355,1255,2147,1487,3610,4785,3261,4712,3022,4252,1751,3742,4344,4091,3752,3239,3371,3884,3154,4449,1619,1370,2132,2414,4836,3514,3682,2015,3889,2464,3810,2433,3198,2663,3578,1951,3600,1819,2200,4187,4746,1210,1079,3926,2112,1740,2958,1423,3271,3166,1837,1602,3499,3349,1888,1296,3877,4402,3535,2181,2211,1407,2344,2313,1128,4461,2228,1573,3799,3731,1089,1804,2895,2841,4108,1227,4636,1540,3119,3078,4012,3440,1397,3968,1360,1350,1762,2100,1270,1851,1772,3122,2990,4512,2685,4789,4544,1856,2694,4525,4693,4845,2509,3102,4233,1672,3704,2724,1719,4227,2460,1329,2323,1292,1182,3714,4750,4545,4530,4676,4291,3141,2877,3094,4167,4191,4513,1700,2613,3032,1810,4861,4648,4054,2108,2038,2551,1207,3723,4867,3555,1644,4814,2472,4621,4180,2317,4736,3242,4384,2304,1456,2851,4482,4540,1874,1979,2140,1627,1825,1324,1114,2932,3486,2125,1099,2505,3819,2813,4347,1429,3409,2052,1599,2697,3364,3920,2272,3201,1759,3755,1318,2926,4644,2761,4625,3183,4476,3856,3306,2076,3999,1466,1185,3675,2634,1434,1857,4184,4567,2208,1695,4765,1187,2440,2408,2793,3508,1167,2645,3422,1648,2677,4613,3890,1483,4393,3839,4269,4365,3707,1727,4470,1754,4116,1103,4594,1219,4412,2941,2524,2376,1154,4662,4341,3768,2938,2225,3811,3519,2188,2514,4032,3458,2716,1124,1256,4505,4041,4144,4283,3255,2393,3086,3272,4122,1795,2398,4590,4769,1428,4485,1669,2326,2293,1312,4218,2935,1107,3594,1332,3487,3700,1258,4526,1139,3796,4796,2308,4645,4099,3330,3832,1765,4854,1897,4640,2425,3416,3091,1464,4433,1356,2311,4790,3996,2642,2383,2774,3419,2097,1492,3710,4300,4374,2197,1175,4716,3678,4498,1282,1894,2987,3038,2422,1117,2257,2284,1722,4805,1144,4586,2116,4616,4073,1801,2856,4667,3138,1571,1376,4509,1667,4430,2632,4541,2610,4569,2111,4452,3192,4726,2974,1112,1877,3747,3055,4009,2129,3668,4064,1993,2780,2267,4028,3977,1838,2785,3006,4652,2874,3483,3971,1344,1556,2542,2329,4684,2710,4554,3480,2962,2350,2212,1528,3110,3917,2288,3953,2244,4304,4626,4067,2069,4844,4832,1198,4720,1841,1213,3838,2835,4502,4373,1643,4085,2369,4362,1726,1364,2390,2984,1973,2506,4473,3009,1948,3444,4136,4260,3293,1340,4355,4722,3571,2005,1413,2914,2816,1583,4458,2665,4868,3672,1420,1081,4177,2629,1990,2202,3455,1472,4688,4113,2564,2597,4581,4812,1551,4749,1748,4035,3842,2661,2192,3214,1605,2732,3129,2175,1230,4045,4658,1798,2298,3737,1770,2863,3315,2977,1611,1225,4272,1743,1780,4754,3058,2386,1531,4324,2747,4314,4561,4240,2700,4549,4768,1235,2276,4351,1085,2701,1130,4409,3246,1790,1987,1716,3940,4190,2165,1813,1300,1637,2688,3202,4222,3623,3412,4781,2044,1285,1738,4164,2354,1934,4802,3652,3829,4133,4436,2320,1504,3961,2892,1272,2848,4150,2675,2056,1404,2411,3488,1702,2379,1221,4488,1353,2279,4510,1955,2145,2860,2722,3943,4225,2584,4671,3284,3400,4289,1839,3452,3565,1459,3533,1844,2194,4559,1216,1126,4102,2305,2658,1998,2626,4138,4770,1966,1157,4419,1834,4656,2347,1654,1289,1453,1734,4310,1807,4061,1276,1326,2594,1476,1674,2554,3336,4126,1745,3630,1854,2262,3613,2707,2290,2251,4666,2135,4681,1508,4715,3104,2004,3003,2764,1822,4834,3516,4705,3859,4261,3085,1491,2475,4197,3267,1440,1308,2680,3993,2902,3645,4619,1161,4745,2447,2648,3585,1777,4387,2066,1077,2362,2162,4869,2917,1994,3552,2374,4794,1625,3264,2522,4843,1673,3132,2142,3379,4125,2258,1133,4472,4399,4687,3860,3511,4670,1678,2983,2719,1097,2587,2604,3464,1380,2008,4841,3496,4491,1757,4579,2768,3690,1480,3015,3956,3328,4350,2704,1410,2366,1710,2404,1891,2619,3309,3822,1657,2334,3347,4286,4157,1693,4865,4442,3765,2668,1165,2021,2063,2159,1829,3025,1268,4242,3869,1976,1357,3020,3538,3886,1880,3394,4088,1093,2353,2502,2241,2127,2600,4382,2485,1537,2539,1713,2484,4691,3812,3341,4674,4584,4784,4547,1490,2087,4858,4655,2640,3950,4779,3227,1367,1944,4702,1945,3212,1681,3405,2018,2866,4161,2832,1848,2059,3052,4863,2993,1449,3721,4174,2980,3035,3575,4552,4002,1417,1870,2771]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05  % Problem  : SWW284+1 : TPTP v9.3.1. Released v5.2.0.
% 0.00/0.13  % Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.25/0.51  % Computer : n004.cluster.edu
% 0.25/0.51  % Model    : x86_64 x86_64
% 0.25/0.51  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.25/0.51  % Memory   : 8046.5625MB
% 0.25/0.51  % OS       : Linux 6.8.0-71-generic
% 0.25/0.51  % CPULimit : 300
% 0.25/0.51  % WCLimit  : 300
% 0.25/0.51  % DateTime : Sat Sep 26 15:35:21 UTC 2026
% 0.25/0.52  % CPUTime  : 
% 0.25/0.52  Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 1.18/1.05  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 3.72/1.82  % [INFO] 	 Parsing done (764ms). 
% 3.72/1.84  % [INFO] 	 Running in sequential loop mode. 
% 4.97/2.42  % [INFO] 	 eprover registered as external prover. 
% 4.97/2.43  % [INFO] 	 Scanning for conjecture ... 
% 5.82/2.76  % [INFO] 	 Found a conjecture (or negated_conjecture) and 1180 axioms. Running axiom selection ... 
% 8.41/3.34  % [INFO] 	 Axiom selection finished. Selected 1074 axioms (removed 106 axioms). 
% 11.68/4.16  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 11.89/4.26  % [INFO] 	 Type checking passed. 
% 11.89/4.26  % [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 ... 
% 52.69/14.65  % External prover 'e' found a proof!
% 52.69/14.65  % [INFO] 	 Killing All external provers ... 
% 52.69/14.65  % Time passed: 13997ms (effective reasoning time: 12800ms)
% 52.69/14.65  % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 52.69/14.68  % Axioms used in derivation (1074): fact_dvd_Oord__less__eq__trans, fact_minus__monom, fact_div__mult__mult1, fact_smult__minus__left, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J, fact_dvd_Oless__trans, fact_mod__div__equality, fact_zle__refl, fact_minus__mult__left, arity_Complex__Ocomplex__Rings_Odvd, fact_add__le__less__mono, fact_Nat_Oadd__0__right, fact_div__add, fact_one__dvd, fact_right__inverse__eq, fact_real__squared__diff__one__factored, arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra, arity_Polynomial__Opoly__Rings_Osemiring__0, arity_Complex__Ocomplex__Rings_Oidom, fact_nat_Osimps_I3_J, fact_coeff__pCons__0, arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, arity_Nat__Onat__Groups_Ocancel__semigroup__add, fact_power__strict__decreasing, arity_Complex__Ocomplex__Groups_Omonoid__add, arity_Int__Oint__Rings_Ocomm__semiring, fact_pdivmod__rel__mult, fact_pos__add__strict, arity_Int__Oint__Rings_Osemiring, fact_mult__poly__0__right, fact_nonzero__inverse__inverse__eq, fact_add__strict__right__mono, fact_zdvd__zmod, fact_div__power, fact_mult__1__left, fact_diff__0__eq__0, fact_dvd__mult__cancel__left, fact_zdvd__not__zless, fact_neg__equal__0__iff__equal, fact_inf__period_I4_J, fact_mod__mult2__eq, fact_diff__add__inverse, fact_dvdI, fact_zdvd__antisym__nonneg, fact_minus__mult__right, arity_Polynomial__Opoly__Rings_Ocomm__semiring__0, fact_odd__nonzero, fact_trans__less__add1, fact_add__poly__code_I2_J, arity_Complex__Ocomplex__Groups_Oab__group__add, fact_inverse__1, fact_pdivmod__rel__unique__div, fact_order, fact_diff__0__right, fact_nat__add__left__commute, fact_mult__nonpos__nonpos, fact_dvd_Oeq__iff, arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors, fact_power__power__power, fact_degree__add__eq__right, fact_mult__left_Ominus, arity_Nat__Onat__Rings_Ono__zero__divisors, fact_nat__less__cases, fact_one__is__add, arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add, fact_dvd_Oorder__trans, fact_zdiff__zmod__right, fact_zminus__zminus, fact_mult__nonneg__nonpos, fact_poly__eq__0__iff__dvd, fact_mult__right__less__imp__less, fact_combine__common__factor, fact_mod__mult__cong, fact_dvd__triv__right, fact_poly__gcd__greatest, fact_n__less__m__mult__n, fact_inverse__inverse__eq, fact_unity__coeff__ex, fact_Suc__mono, fact_less__nat__zero__code, fact_add__less__mono, fact_zadd__commute, fact_mult__pCons__right, fact_mult__eq__0__iff, fact_pow__divides__pow__int, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J, fact_smult__dvd__cancel, fact_split__mult__neg__le, fact_zpower__zmod, fact_dvd_Oless__asym_H, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J, fact_Suc__times__mod__eq, fact_inverse__add, fact_zero__less__power, arity_Nat__Onat__Rings_Ocomm__semiring__0, fact_psize__eq__0__iff, fact_nat_Osize_I4_J, arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le, arity_Nat__Onat__Groups_Omonoid__mult, fact_power__increasing, fact_diff__Suc__less, fact_divide__inverse, fact_mult__0, fact_inverse__zero__imp__zero, fact_mod__mult__eq, fact_dvd_Oless__imp__not__less, fact_zero__less__diff, fact_nonzero__inverse__eq__imp__eq, fact_mult__mono, fact_nat__add__right__cancel, fact_field__power__not__zero, fact_dvd__smult, fact_mult__poly__0__left, arity_Nat__Onat__Rings_Ozero__neq__one, fact_poly__zero, fact_eq__neg__iff__add__eq__0, arity_Polynomial__Opoly__Groups_Oab__semigroup__mult, fact_add__right__imp__eq, fact_realpow__two__diff, arity_Complex__Ocomplex__Groups_Oab__semigroup__mult, arity_Polynomial__Opoly__Rings_Oidom, fact_dvd__mod, fact_gr0__conv__Suc, fact_coeff__eq__0, fact_poly__dvd__antisym, fact_zdvd__mult__cancel, fact_inverse__zero, fact_dvd_Oord__eq__less__trans, fact_zmod__zdiv__equality, fact_Suc__diff__diff, fact_mod__diff__cong, fact_fundamental__theorem__of__algebra, fact_minus__add__cancel, fact_smult__smult, arity_Polynomial__Opoly__Rings_Ocomm__ring, fact_zle__linear, fact_nat__add__left__cancel__less, fact_mod__mult__distrib2, fact_coeff__inverse, fact_mod__less, fact_coeff__add, fact_mult_Oadd__right, fact_mult_Ominus__right, help_c__fequal__2, fact_mult__eq__self__implies__10, fact_minus__pCons, fact_poly__mod__minus__left, fact_dvd__div__mult__self, fact_degree__linear__power, fact_pow__divides__pow__nat, fact_zero__le__one, fact_zmult__div__cancel, fact_poly__add, fact_degree__pCons__0, fact_pdivmod__rel, fact_order__1, fact_int__0__less__1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J, help_c__fFalse__1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J, fact_linorder__neqE__linordered__idom, arity_Complex__Ocomplex__Rings_Ocomm__semiring__1, fact_synthetic__div__correct_H, fact_smult__1__left, fact_zadd__left__mono, fact_mult__right_Ozero, fact_add__le__imp__le__left, fact_mod__mult__self1, fact_sum__squares__ge__zero, fact_pow__divides__eq__nat, fact_pdivmod__rel__by__0, arity_Complex__Ocomplex__Rings_Odivision__ring__inverse__zero, fact_mult__smult__right, fact_smult__pCons, fact_mod__if, fact_div__1, fact_mod__by__1, arity_Polynomial__Opoly__Rings_Olinordered__semidom, fact_divide__1, fact_mod__mult__self4, fact_zdvd__mult__div__cancel, fact_dvd__reduce, fact_synthetic__div__0, arity_Int__Oint__Groups_Ozero, fact_zdiv__zero, fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J, fact_mult_Ozero__left, fact_zdvd__zmod__imp__zdvd, fact_even__less__0__iff, fact_power__commutes, fact_add__mono, fact_add__0__iff, fact_poly__smult, fact_degree__1, arity_Int__Oint__Groups_Ocancel__comm__monoid__add, fact_power__strict__increasing, fact_power__one__over, arity_Nat__Onat__Rings_Osemiring__0, fact_one__le__power, arity_Int__Oint__Groups_Ocomm__monoid__add, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J, fact_add__monom, fact_add__0__left, fact_zmod__zminus1__eq__if, fact_poly__0, fact_gr0I, fact_sgn__minus, fact_add__eq__if, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J, fact_nullstellensatz__lemma, fact_coeff__1, arity_Int__Oint__Rings_Olinordered__ring, fact_add__right__mono, fact_Divides_Otransfer__nat__int__function__closures_I2_J, fact_minus__add, fact_divide__eq__imp, arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add, fact_power__minus, fact_one__less__power, fact_mod__mult__left__eq, fact_nat__dvd__not__less, fact_leading__coeff__neq__0, fact_mult__eq__1__iff, fact_dvd__div__eq__mult, fact_smult__add__right, fact_degree__pCons__eq__if, arity_Nat__Onat__Rings_Oordered__comm__semiring, fact_power__eq__0__iff, fact_diff__Suc__eq__diff__pred, fact_divide__add__eq__iff, fact_less__add__eq__less, fact_nat__mult__1__right, fact_mod__1, arity_Complex__Ocomplex__Rings_Osemiring__0, fact_poly__gcd__monic, fact_dvd__eq__mod__eq__0, fact_coeff__linear__power, fact_poly__gcd_Oassoc, fact_zdiv__zmult1__eq, fact_pos__imp__zdiv__neg__iff, fact_mult__right_Odiff, fact_one__reorient, fact_diff__commute, fact_minus__mult__commute, arity_Polynomial__Opoly__Groups_Omonoid__add, fact_division__ring__inverse__add, fact_mult__left_Odiff, fact_eq__iff__diff__eq__0, fact_add__lessD1, fact_div__mult__self1__is__id, arity_Int__Oint__Rings_Oidom, arity_Complex__Ocomplex__Fields_Ofield, arity_Polynomial__Opoly__Int_Oring__char__0, fact_lessI, fact_zero__le__square, fact_dvd_Oantisym__conv, fact_zero__reorient, fact_dvd__mult__cancel2, fact_dvd__0__right, fact_mod__add__self1, arity_Int__Oint__Rings_Oordered__cancel__semiring, fact_minus__divide__divide, fact_zadd__zless__mono, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J, fact_power__gt1__lemma, fact_dvd_Oord__eq__le__trans, arity_Polynomial__Opoly__Rings_Oring__1, fact_mult__less__mono1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J, fact_poly__mult, fact_split__mult__pos__le, fact_offset__poly__pCons, fact_nat_Oinject, fact_neg__mod__bound, fact_zmod__zminus2__eq__if, arity_Polynomial__Opoly__Power_Opower, arity_Int__Oint__Rings_Oring__no__zero__divisors, fact_not__add__less2, fact_dvd__trans, fact_mod__mod__trivial, fact_int__div__less__self, fact_Suc__less__SucD, fact_dvd_Oeq__refl, arity_Polynomial__Opoly__Groups_Osgn__if, fact_sgn__zero, fact_field__divide__inverse, fact_synthetic__div__pCons, fact_zadd__strict__right__mono, arity_Complex__Ocomplex__RealVector_Oreal__field, fact_zdiv__zadd1__eq, fact_zdvd__reduce, fact_add__0__right, fact_sgn__mult, fact_diff__less, fact_div__mult__self2, fact_power__Suc2, fact_div__mult2__eq, fact_DERIV__inverse__lemma, fact_minus__divide__right, fact_zadd__assoc, arity_Polynomial__Opoly__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J, fact_power__inverse, fact_less__irrefl__nat, fact_mod__smult__right, fact_mult_Oprod__diff__prod, fact_mod__diff__right__eq, fact_Suc__lessD, fact_pdivmod__rel__0__iff, fact_not__less__less__Suc__eq, fact_diff__def, fact_dvd__mult2, fact_nat__mult__1, fact_mult__right__mono__neg, fact_diff__pCons, fact_mod__div__equality2, fact_zpower__zadd__distrib, fact_dvd_Oless__imp__neq, fact_diff__self, fact_minus__nat_Odiff__0, fact_pos__zmult__eq__1__iff, arity_Int__Oint__Groups_Ocancel__semigroup__add, fact_nonzero__inverse__mult__distrib, fact_realpow__minus__mult, fact_offset__poly__single, arity_Int__Oint__Power_Opower, fact_diff__0, fact_add__left__imp__eq, fact_poly__gcd__minus__right, arity_Nat__Onat__Groups_Ocancel__comm__monoid__add, fact_zpower__zpower, fact_pcompose__0, fact_mult__mono_H, fact_dvd__pos__nat, arity_Nat__Onat__Power_Opower, fact_poly__gcd__dvd2, fact_sgn__less, fact_ab__semigroup__mult__class_Omult__ac_I1_J, fact_not__less0, fact_mod__mult__self1__is__0, arity_Int__Oint__Rings_Osemiring__0, fact_monom__Suc, fact_trans__less__add2, fact_poly__offset__poly, fact_nat__diff__split, fact_zadd__0, fact_add__frac__eq, fact_mod__less__divisor, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J, arity_Complex__Ocomplex__Groups_Oab__semigroup__add, arity_Complex__Ocomplex__Fields_Ofield__inverse__zero, fact_dvd_Oless__asym, fact_offset__poly__0, fact_zmod__zdiv__trivial, fact_inverse__divide, arity_Nat__Onat__Rings_Ocomm__semiring__1, fact_coeff__minus, fact_mod__pCons, fact_synthetic__div__correct, fact_degree__smult__eq, fact_not__less__eq, arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra, fact_nat_Osimps_I2_J, fact_mult__divide__mult__cancel__left, fact_zero__le__double__add__iff__zero__le__single__add, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J, arity_Int__Oint__Groups_Oab__semigroup__mult, fact_degree__pCons__eq, fact_pCons__0__0, fact_divide_Ominus, fact_zmod__simps_I3_J, arity_Complex__Ocomplex__Rings_Osemiring, fact_pdivmod__rel__smult__right, arity_Int__Oint__Divides_Osemiring__div, fact_zmod__zero, fact_smult__diff__right, fact_less__zeroE, fact_termination__basic__simps_I2_J, fact_minus__zero, arity_Complex__Ocomplex__Groups_Ogroup__add, fact_not__one__less__zero, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J, fact_poly__replicate__append, fact_nat__neq__iff, fact_nat__1__eq__mult__iff, fact_nonzero__inverse__eq__divide, fact_offset__poly__eq__0__iff, fact_mult__left__less__imp__less, fact_diff__mult__distrib2, fact_dvd_Ole__less, fact_sgn__poly__def, fact_smult__eq__0__iff, arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict, fact_add__mult__distrib2, arity_Int__Oint__Groups_Ocancel__ab__semigroup__add, fact_Limits_Ominus__diff__minus, fact_less__not__refl3, arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult, fact_add__right__cancel, fact_one__less__mult, fact_dvd__mult, fact_add__left__cancel, fact_poly__gcd__minus__left, arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add, fact_add__pCons, fact_neg__less__nonneg, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J, arity_Complex__Ocomplex__RealVector_Oreal__normed__field, fact_poly__div__mult__right, fact_eq__add__iff1, arity_Complex__Ocomplex__Rings_Ozero__neq__one, fact_sgn__sgn, fact_mod__Suc, fact_poly__pCons, fact_nat__diff__split__asm, fact_zle__antisym, fact_nat__mult__dvd__cancel__disj, fact_dvd_Oorder__refl, fact_mult__nonneg__nonneg, fact_add__less__cancel__left, fact_poly__rec__pCons, arity_Nat__Onat__Divides_Osemiring__div, fact_frac__eq__eq, fact_zmult__assoc, fact_nat__size, arity_Complex__Ocomplex__Rings_Ocomm__semiring, arity_Int__Oint__Rings_Oordered__semiring, fact_dvd_Oord__le__eq__trans, fact_add__divide__eq__iff, fact_div__by__1, fact_bool_Osize_I2_J, fact_neg__0__equal__iff__equal, fact_zminus__zadd__distrib, arity_Polynomial__Opoly__Rings_Oordered__comm__semiring, fact_mult__left_Oadd, fact_double__zero__sym, fact_mod__mult__mult1, fact_pdivmod__rel__smult__left, fact_zmult__1, fact_mult__left_Ozero, fact_one__neq__zero, fact_div__less, fact_left__minus, fact_diff__cancel, fact_div__dvd__div, fact_right__minus, fact_mult__left__mono, fact_mod__div__trivial, fact_power__decreasing, fact_monom__eq__0__iff, fact_split__mod, fact_mult__dvd__mono, fact_poly__1, fact_less__Suc__eq__0__disj, fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J, fact_dvd__smult__cancel, fact_less__add__iff2, arity_Polynomial__Opoly__Rings_Oring, fact_add__strict__mono, fact_diff__divide__eq__iff, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J, fact_offset__poly__eq__0__lemma, arity_Polynomial__Opoly__Groups_Oab__group__add, fact_poly__pcompose, arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add, arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult, fact_add__num__frac, fact_mult__cancel2, arity_Int__Oint__Rings_Ocomm__ring, fact_degree__offset__poly, fact_nat__mult__assoc, arity_Complex__Ocomplex__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_equation__minus__iff, fact_inverse__eq__divide, fact_poly__gcd__dvd1, fact_diff__int__def__symmetric, arity_Nat__Onat__Groups_Ocomm__monoid__add, fact_mod__add__cong, fact_Suc__mult__less__cancel1, arity_Polynomial__Opoly__Groups_Oab__semigroup__add, fact_less__antisym, fact_not__one__le__zero, arity_Polynomial__Opoly__Rings_Olinordered__idom, arity_Int__Oint__Groups_Ogroup__add, fact_add__diff__cancel, fact_div__0, fact_poly__minus, arity_Polynomial__Opoly__Rings_Oordered__semiring, arity_Nat__Onat__Rings_Ocomm__semiring, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J, arity_Int__Oint__Rings_Olinordered__semidom, fact_zmod__zdiv__equality_H, fact_int__power__div__base, fact_equal__neg__zero, fact_div__mult__div__if__dvd, fact_div__mult__self1__is__m, fact_mult_Ominus__left, help_c__fequal__1, arity_Complex__Ocomplex__Rings_Oring, fact_nat_Osize_I1_J, fact_zmult__commute, fact_comm__mult__strict__left__mono, fact_smult__add__left, fact_diff__int__def, arity_Nat__Onat__Groups_Omonoid__add, fact_monom__0, fact_power__Suc__less__one, fact_mult__monom, fact_pos__poly__mult, fact_not__sum__squares__lt__zero, fact_order__2, fact_divide__eq__eq, fact_mult__nonneg__nonpos2, fact_less__add__Suc2, arity_Polynomial__Opoly__Groups_Omonoid__mult, fact_dvd__power__same, fact_less__imp__diff__less, fact_div__add__self1, fact_power__add, fact_inverse__eq__imp__eq, fact_pdivmod__rel__unique, fact_nat__mult__eq__cancel1, arity_Polynomial__Opoly__Divides_Osemiring__div, fact_sgn__0__0, arity_Int__Oint__Rings_Oring__1__no__zero__divisors, fact_Suc__pred_H, fact_pCons__eq__iff, fact_dvd_Oless__not__sym, fact_degree__add__less, fact_mod__by__0, fact_diff__add__cancel, fact_zdiv__zmod__equality2, arity_Int__Oint__Groups_Oab__group__add, fact_pdivmod__rel__by__0__iff, fact_minus__minus, fact_dvd__mult__cancel1, fact_mod__diff__eq, fact_bool_Osize_I1_J, fact_smult__0__left, fact_minus__add__distrib, fact_diffs0__imp__equal, fact_nonzero__divide__eq__eq, arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add, arity_Nat__Onat__Groups_Ozero, fact_pCons__eq__0__iff, arity_Complex__Ocomplex__Groups_Oone, fact_add__is__1, arity_Nat__Onat__Groups_Oordered__ab__semigroup__add, fact_inverse__unique, fact_dvd__refl, fact_zero__less__power__nat__eq, fact_power_Opower_Opower__0, fact_power__Suc__less, fact_mod__mult__right__eq, fact_neg__imp__zdiv__neg__iff, fact_linorder__neqE__nat, fact_mult__less__mono2, fact_smult__monom, fact_nat__mult__div__cancel__disj, fact_zmod__zminus__zminus, arity_Complex__Ocomplex__Groups_Ocomm__monoid__add, fact_dvd__diff, fact_diff__diff__left, fact_power__inject__exp, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J, fact_nat__mult__less__cancel1, fact_dvd__div__neg, fact_zmod__eq__dvd__iff, fact_dvd__add, arity_Complex__Ocomplex__Rings_Oring__1, arity_Complex__Ocomplex__Rings_Odivision__ring, fact_Suc__mult__cancel1, arity_Nat__Onat__Groups_Ocomm__monoid__mult, arity_Int__Oint__Groups_Omonoid__mult, fact_mult__poly__add__left, fact_mult_Ozero__right, fact_zdiv__zmult2__eq, fact_power__divide, arity_Int__Oint__Int_Oring__char__0, fact_sgn__times, fact_zless__le, fact_zadd__left__commute, fact_dvd__neg__div, fact_coeff__pCons, fact_one__poly__def, fact_mod__minus__eq, fact_order__root, fact_mod__mult__self2__is__0, fact_power__le__imp__le__exp, fact_power__one__right, fact_dvd_Oneq__le__trans, fact_pos__mod__bound, fact_less__poly__def, fact_add__less__cancel__right, fact_not__add__less1, fact_div__add1__eq, fact_inverse__nonzero__iff__nonzero, fact_mod__add__self2, fact_div__mult__self1, fact_div__mod__equality2, fact_mod__geq, fact_power__increasing__iff, fact_expand__poly__eq, fact_mult__right_Oadd, fact_ext, fact_mod__self, fact_poly__gcd_Oleft__commute, arity_Polynomial__Opoly__Rings_Ocomm__semiring__1, fact_poly__gcd_Ocommute, fact_mult_Oadd__left, arity_Complex__Ocomplex__Rings_Ocomm__ring__1, fact_zero__le__power, fact_Suc__neq__Zero, fact_power__strict__increasing__iff, arity_Polynomial__Opoly__Rings_Omult__zero, fact_inverse__eq__1__iff, arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors, fact_Suc__inject, fact_dvd__poly__gcd__iff, fact_minus__dvd__iff, fact_less__diff__conv, fact_diff__poly__code_I2_J, fact_crossproduct__noteq, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J, arity_Nat__Onat__Rings_Olinordered__semiring, fact_zle__trans, fact_less__add__iff1, fact_div__mult__mult1__if, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J, fact_division__ring__inverse__diff, fact_zero__less__double__add__iff__zero__less__single__add, fact_nat__power__less__imp__less, fact_nat__lt__two__imp__zero__or__one, fact_times__divide__eq__right, fact_div__mult__mult2, fact_poly__gcd__code, fact_eq__divide__imp, fact_add__eq__self__zero, fact_diff__less__Suc, fact_mod__poly__less, fact_zmod__eq__0__iff, arity_Int__Oint__Rings_Omult__zero, fact_zmod__le__nonneg__dividend, fact_div__less__dividend, fact_dvd_Oless__imp__le, fact_dvd__div__div__eq__mult, fact_nat__mult__dvd__cancel1, fact_dvd__div__mult, fact_dvd__mult__div__cancel, fact_divide__self__if, fact_mult__is__0, fact_nat__add__left__cancel, fact_Suc__diff__1, fact_uminus__dvd__conv_I2_J, fact_divide_Ozero, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J, fact_zdiv__zminus2__eq__if, fact_n__not__Suc__n, fact_pe, fact_zminus__zmod, fact_mult__zero__right, fact_mod__add__left__eq, arity_Int__Oint__Rings_Oring, fact_divide__zero, fact_div__by__0, fact_zminus__0, fact_poly__rec__0, fact_mod__diff__left__eq, fact_div__smult__left, fact_power__less__imp__less__exp, arity_Nat__Onat__Rings_Omult__zero, fact_zdiv__zmod__equality, fact_inverse__mult__distrib, fact_less__minus__self__iff, fact_pos__poly__total, arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict, fact_add__strict__left__mono, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J, fact_zmod__simps_I4_J, fact_mult__less__cancel1, fact_nat__0__less__mult__iff, fact_add__gr__0, arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add, fact_zdiff__zmult__distrib2, fact_inverse__eq__iff__eq, fact_dvd__1__left, arity_Polynomial__Opoly__Rings_Ozero__neq__one, fact_add__mult__distrib, fact_diff__poly__code_I1_J, fact_add__Suc, fact_dvd__triv__left, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J, fact_diff__divide__distrib, fact_coeff__mult__degree__sum, fact_mult__divide__mult__cancel__right, fact_nat_Osize_I2_J, fact_minus__unique, fact_poly__power, fact_poly__gcd__1__right, fact_zdiv__zminus1__eq__if, fact_diff__less__mono2, fact_dvd_Oless__imp__not__eq, fact_divide__zero__left, fact_diff__mult__distrib, fact_int__0__neq__1, fact_power__Suc, fact_add__less__imp__less__right, fact_zdiff__zmult__distrib, fact_Suc__eq__plus1__left, fact_zdiff__zmod__left, arity_Complex__Ocomplex__Rings_Ono__zero__divisors, arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add, fact_left__inverse, fact_power__0, fact_zero__less__one, fact_power__0__left, fact_add_Ocomm__neutral, fact_div__add__self2, arity_Nat__Onat__Rings_Oordered__cancel__semiring, fact_poly__div__minus__left, fact_dvd__diff__nat, arity_Nat__Onat__Rings_Odvd, arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le, fact_zmod__zminus2__not__zero, fact_degree__synthetic__div, fact_Zero__not__Suc, fact_dvd__mult__right, fact_gcd__lcm__complete__lattice__nat_Otop__greatest, fact_neg__equal__zero, fact_nonzero__inverse__minus__eq, fact_power__mult, fact_add__minus__cancel, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J, fact_nonzero__imp__inverse__nonzero, arity_Int__Oint__Groups_Osgn__if, fact_constant__def, arity_Nat__Onat__Rings_Oordered__semiring, fact_mult__Suc, fact_less__trans__Suc, fact_le__minus__self__iff, fact_semiring__div__class_Omod__div__equality_H, fact_nat__add__assoc, fact_mod__mult__self3, fact_diff__frac__eq, fact_poly__rec_Osimps, fact_poly__gcd__zero__iff, fact_mult__cancel1, arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors, fact_poly__monom, fact_psize__def, fact_zmod__self, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J, fact_mult__zero__left, arity_Int__Oint__Groups_Oordered__ab__semigroup__add, fact_left__add__mult__distrib, fact_add__scale__eq__noteq, fact_eq__add__iff2, fact_zadd__zmult__distrib2, fact_less__not__refl2, fact_field__class_Onormalizing__field__rules_I2_J, fact_zdvd__mono, fact_monom__eq__0, fact_zmod__zmult1__eq, fact_power__eq__if, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J, fact_leading__coeff__0__iff, fact_less__add__Suc1, fact_nat__case__Suc, fact_One__nat__def, fact_div__mult__self__is__m, fact_divide__diff__eq__iff, fact_mult__diff__mult, fact_realpow__num__eq__if, fact_add__less__imp__less__left, fact_times__divide__times__eq, fact_divide_Oadd, fact_ab__diff__minus, fact_sgn__zero__iff, arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add, arity_Int__Oint__Rings_Olinordered__semiring, fact_coeff__smult, fact_nonzero__minus__divide__divide, fact_smult__0__right, fact_ab__left__minus, fact_sgn0, fact_dvd__mod__iff, fact_minus__diff__eq, fact_comm__semiring__class_Odistrib, fact_square__eq__iff, fact_smult__dvd__iff, fact_nonzero__minus__divide__right, fact_add__is__0, fact_mod__eq__0__iff, arity_Int__Oint__Rings_Oordered__ring, fact_mod__mult__mult2, fact_power_Opower_Opower__Suc, fact_Deriv_Oadd__diff__add, fact_mult__right__mono, fact_pCons__def, arity_Polynomial__Opoly__Rings_Oordered__ring, fact_diff__minus__eq__add, fact_power__mono, fact_Suc__eq__plus1, fact_diff__self__eq__0, fact_add__eq__0__iff, fact_minus__poly__code_I2_J, fact_Divides_Otransfer__nat__int__function__closures_I1_J, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J, fact_neg__equal__iff__equal, fact_degree__minus, fact_pos__poly__add, fact_nat__mult__div__cancel1, fact_diff__add__0, fact_uminus__dvd__conv_I1_J, fact_dvd__minus__iff, fact_nat__case__0, arity_Nat__Onat__Groups_Oab__semigroup__mult, fact_diff__cancel2, fact_add__0, fact_poly__mod__minus__right, fact_dvd__mult__cancel, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J, arity_Int__Oint__Rings_Olinordered__idom, arity_Int__Oint__Rings_Ocomm__semiring__1, fact_sgn__greater, fact_realpow__two__disj, fact_inverse__minus__eq, fact_div__neg__pos__less0, fact_coeff__pCons__Suc, fact_smult__dvd, arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1, fact_mult__nonpos__nonneg, fact_zmod__simps_I2_J, fact_pos__poly__pCons, arity_Polynomial__Opoly__Rings_Osemiring, fact_Zero__neq__Suc, fact_poly__gcd__0__0, fact_degree__add__eq__left, arity_Nat__Onat__Groups_Oab__semigroup__add, fact_add__left__mono, arity_Int__Oint__Rings_Oordered__comm__semiring, fact_mod__smult__left, fact_mod__mod__cancel, fact_zmult__zless__mono2, fact_Suc__pred, arity_Int__Oint__Groups_Oab__semigroup__add, fact_inf__period_I3_J, fact_nat__dvd__1__iff__1, fact_zmod__simps_I1_J, arity_Int__Oint__Groups_Oone, arity_Nat__Onat__Rings_Olinordered__semidom, fact_add__le__cancel__right, arity_Complex__Ocomplex__Groups_Omonoid__mult, arity_Int__Oint__Groups_Ocomm__monoid__mult, fact_power__Suc__0, fact_nat__mult__eq__1__iff, fact_n__less__n__mult__m, fact_mult__right_Ominus, fact_less__not__refl, fact_double__add__le__zero__iff__single__add__le__zero, fact_poly__gcd__unique, fact_synthetic__div__eq__0__iff, fact_power__mult__distrib, fact_less__1__mult, fact_right__inverse, fact_coeff__monom, fact_power__one, fact_zmod__zmult2__eq, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J, fact_poly__diff, fact_minus__mult__minus, fact_crossproduct__eq, fact_monom__eq__iff, fact_square__eq__1__iff, fact_dvd_Oantisym, fact_comm__mult__left__mono, fact_degree__mod__less, fact_gr__implies__not0, fact_zdiv__zminus__zminus, fact_dvd_Ole__neq__trans, fact_add__less__le__mono, arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add, fact_power__less__power__Suc, arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors, fact_less__add__one, fact_degree__monom__eq, fact_less__Suc__eq, fact_nonzero__power__inverse, fact_zmod__zminus2, fact_dvd__smult__iff, fact_diff__Suc__1, fact_dvd__antisym, fact_div__mod__equality, fact_poly__gcd_Osimps_I1_J, fact_zero__neq__one, fact_nat__add__commute, fact_mult__smult__left, fact_add__poly__code_I1_J, fact_synthetic__div__unique, arity_Int__Oint__Rings_Ono__zero__divisors, fact_Suc__less__eq, fact_degree__diff__less, arity_Int__Oint__Rings_Oring__1, fact_right__minus__eq, fact_div__mult__swap, arity_Int__Oint__Rings_Odvd, fact_poly__gcd__1__left, fact_mult_Odiff__left, fact_coeff__0, arity_Polynomial__Opoly__Groups_Ogroup__add, fact_zmult__zminus, fact_sgn__one, arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add, fact_zdiv__zminus2, fact_zero__less__Suc, fact_mod__poly__eq, arity_Polynomial__Opoly__Rings_Olinordered__semiring, help_c__fTrue__1, fact_div__self, fact_plus__nat_Oadd__0, arity_Complex__Ocomplex__Rings_Ocomm__ring, fact_mult_Ocomm__neutral, fact_nat_Osize_I3_J, fact_dvd_Oless__le, fact_not__pos__poly__0, fact_dvd_Oless__le__trans, fact_nat__mult__commute, fact_nat__power__eq__Suc__0__iff, fact_divide_Odiff, arity_Polynomial__Opoly__Divides_Oring__div, fact_mod__Suc__eq__Suc__mod, fact_diff__add__inverse2, arity_Complex__Ocomplex__Int_Oring__char__0, fact_diff__eq__diff__eq, fact_minus__le__self__iff, fact_zdiv__self, fact_pow__divides__eq__int, fact_neq0__conv, fact_Suc__n__not__n, fact_mult__left__mono__neg, fact_mult__0__right, fact_zless__linear, fact_add__divide__distrib, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J, arity_Int__Oint__Rings_Olinordered__comm__semiring__strict, fact_div__poly__eq, arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring, arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le, arity_Polynomial__Opoly__Rings_Ocomm__ring__1, fact_add__frac__num, fact_minus__equation__iff, fact_synthetic__div__unique__lemma, fact_zdvd__period, fact_split__neg__lemma, fact_zadd__0__right, arity_Polynomial__Opoly__Rings_Odvd, arity_Polynomial__Opoly__Rings_Ocomm__semiring, arity_Complex__Ocomplex__Rings_Omult__zero, fact_zmult__1__right, arity_Polynomial__Opoly__Rings_Ono__zero__divisors, fact_double__eq__0__iff, fact_mult__Suc__right, fact_zdvd__zdiffD, fact_dvd__0__left, fact_pcompose__pCons, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J, fact_mult__less__cancel2, fact_mult__1__right, fact_zero__less__two, fact_gcd__lcm__complete__lattice__nat_Obot__least, fact_poly__div__minus__right, arity_Polynomial__Opoly__Groups_Ozero, conj_0, fact_dvd__iff__poly__eq__0, arity_Int__Oint__Rings_Ocomm__ring__1, fact_add__diff__inverse, arity_Nat__Onat__Rings_Osemiring, fact_smult__minus__right, fact_termination__basic__simps_I1_J, fact_less__iff__Suc__add, fact_dvd__1__iff__1, fact_DIVISION__BY__ZERO, fact_mod__lemma, fact_mod__0, fact_add__le__imp__le__right, fact_pdivmod__rel__unique__mod, fact_nat__zero__less__power__iff, fact_mod__add__right__eq, fact_less__Suc0, fact_div__mult1__eq, fact_pos__poly__def, fact_dvd_Ole__less__trans, fact_diff__monom, arity_Int__Oint__Groups_Olinordered__ab__group__add, fact_nat__mult__eq__cancel__disj, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J, arity_Int__Oint__Rings_Ozero__neq__one, fact_add__imp__eq, fact_power__gt1, fact_mult__eq__if, fact_zadd__zmult__distrib, arity_Polynomial__Opoly__Rings_Olinordered__ring, arity_Polynomial__Opoly__Groups_Ocomm__monoid__add, fact_degree__mult__eq, arity_Complex__Ocomplex__Power_Opower, fact_zadd__zminus__inverse2, fact_div__mult__self2__is__id, fact_DERIV__mult__lemma, fact_ab__semigroup__add__class_Oadd__ac_I1_J, arity_Nat__Onat__Groups_Oone, fact_mod__mult__self2, fact_diff__Suc__Suc, fact_dvd__mult__left, fact_coeff__inject, arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_degree__0, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J, fact_mod__minus__cong, fact_dvd_Ole__imp__less__or__eq, fact_power__0__Suc, fact_minus__poly__code_I1_J, arity_Complex__Ocomplex__RealVector_Oreal__normed__vector, fact_no__zero__divisors, fact_Deriv_Oinverse__diff__inverse, fact_less__SucE, fact_zless__add1__eq, fact_add__less__mono1, fact_add__le__cancel__left, fact_eq__divide__eq, fact_divisors__zero, fact_smult__diff__left, arity_Polynomial__Opoly__Groups_Oone, fact_poly__eq__iff, fact_field__inverse__zero, fact_nonzero__eq__divide__eq, fact_double__add__less__zero__iff__single__add__less__zero, fact_minus__divide__left, fact_zmod__zminus1__not__zero, fact_pdivmod__rel__0, fact_odd__less__0, fact_dvd_Oless__imp__not__eq2, fact_pdivmod__rel__pCons, fact_field__inverse, arity_Int__Oint__Groups_Omonoid__add, arity_Int__Oint__Rings_Ocomm__semiring__0, fact_add__Suc__shift, fact_dvd__mult__cancel__right, fact_mult__1, fact_pdivmod__rel__def, fact_mod__add__eq, arity_Int__Oint__Divides_Oring__div, fact_dvd__imp__mod__0, fact_poly__decompose, fact_less__SucI, fact_Suc__not__Zero, fact_dvd__power, arity_Complex__Ocomplex__Groups_Ozero, fact_poly__gcd_Osimps_I2_J, arity_Complex__Ocomplex__Groups_Ocancel__comm__monoid__add, fact_not__square__less__zero, fact_Suc__lessI, fact_add__Suc__right, fact_mult__pCons__left, fact_less__bin__lemma, arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add, fact_mult_Odiff__right, fact_mod__mult__distrib, arity_Complex__Ocomplex__Rings_Ocomm__semiring__0, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J, fact_divide__self, fact_nonzero__power__divide, fact_coeff__diff, fact_dvd__mod__imp__dvd
% 52.69/14.68  % No. of inferences in proof: 2152
% 52.69/14.68  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 13997 ms resp. 12800 ms w/o parsing
% 58.31/15.77  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 58.31/15.79  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------