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

%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : SWW186+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 : n009.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 09:09:36 AM UTC 2026

% Result   : Theorem 32.56s 14.40s
% Output   : Refutation 36.93s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      : 1006
% Syntax   : Number of formulae    : 2014 ( 407 unt;   0 typ;   0 def)
%            Number of atoms       : 5360 (1724 equ;   0 cnn)
%            Maximal formula atoms :   15 (   2 avg)
%            Number of connectives : 28175 ( 355   ~; 129   |; 417   &;24474   @)
%                                         ( 182 <=>;2618  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   8 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  100 (  98 usr;   7 con; 0-5 aty)
%            Number of variables   : 4977 (   0   ^;4946   !;  31   ?;4977   :)

% Comments : 
%------------------------------------------------------------------------------
thf(v_a_decl,type,
    v_a: $i ).

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

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

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

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

thf(class_Fields_Olinordered__field__inverse__zero_decl,type,
    class_Fields_Olinordered__field__inverse__zero: $i > $o ).

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

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

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

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

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

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

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

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

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

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

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

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_Groups_Ogroup__add_decl,type,
    class_Groups_Ogroup__add: $i > $o ).

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

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

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

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_Rings_Oordered__ring_decl,type,
    class_Rings_Oordered__ring: $i > $o ).

thf(class_Groups_Ouminus_decl,type,
    class_Groups_Ouminus: $i > $o ).

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

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

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

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

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

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

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

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_Divides_Oring__div_decl,type,
    class_Divides_Oring__div: $i > $o ).

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

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

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

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

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

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_Rings_Oring_decl,type,
    class_Rings_Oring: $i > $o ).

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

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

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

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

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

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

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

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_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_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_Rings_Ono__zero__divisors_decl,type,
    class_Rings_Ono__zero__divisors: $i > $o ).

thf(class_Groups_Oplus_decl,type,
    class_Groups_Oplus: $i > $o ).

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

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

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

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

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

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

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

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

thf(v_h_decl,type,
    v_h: $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(373,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(2199,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)],[373]) ).

thf(837,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(3883,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)],[837]) ).

thf(198,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(1633,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)],[198]) ).

thf(812,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(3798,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)],[812]) ).

thf(773,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(3666,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)],[773]) ).

thf(76,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(1237,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)],[76]) ).

thf(278,axiom,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le0) ).

thf(1899,plain,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(257,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(1840,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)],[257]) ).

thf(29,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(1083,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)],[29]) ).

thf(412,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(2331,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)],[412]) ).

thf(295,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(1950,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)],[295]) ).

thf(387,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__1) ).

thf(2250,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(562,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(2822,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)],[562]) ).

thf(911,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(4188,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)],[911]) ).

thf(914,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(4201,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)],[914]) ).

thf(451,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(2452,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)],[451]) ).

thf(991,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(4447,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)],[991]) ).

thf(268,axiom,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__mult) ).

thf(1879,plain,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(340,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(2094,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)],[340]) ).

thf(134,axiom,
    tc_Int_Oint @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Omult__zero) ).

thf(1411,plain,
    tc_Int_Oint @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(502,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(2612,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)],[502]) ).

thf(150,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring) ).

thf(1469,plain,
    tc_Int_Oint @ class_Rings_Olinordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(619,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(3053,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)],[619]) ).

thf(884,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(4101,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)],[884]) ).

thf(931,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(4262,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Power_Opower ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(233,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mult__le__cancel1) ).

thf(1767,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(390,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__diff__iff) ).

thf(2263,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
          & ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(850,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(3930,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)],[850]) ).

thf(586,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1 )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convex__bound__le) ).

thf(2921,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1 )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(782,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__0) ).

thf(3698,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(101,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( 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__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdvd__imp__le) ).

thf(1315,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( 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__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(823,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(3830,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)],[823]) ).

thf(552,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(2787,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)],[552]) ).

thf(394,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__less__eq__eq) ).

thf(2280,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(39,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(1110,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)],[39]) ).

thf(649,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(3172,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)],[649]) ).

thf(870,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(4037,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)],[870]) ).

thf(287,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1__strict )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convex__bound__lt) ).

thf(1924,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F @ class_Rings_Olinordered__semiring__1__strict )
     => ( ( D @ ( E @ ( F @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( D @ ( C @ ( F @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( B @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( A @ ( F @ c_Groups_Ozero__class_Ozero @ ( F @ c_Orderings_Oord__class_Oless__eq ) ) )
             => ( ( ( A @ ( B @ ( F @ c_Groups_Oplus__class_Oplus ) ) )
                  = ( F @ c_Groups_Oone__class_Oone ) )
               => ( D @ ( C @ ( A @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( E @ ( B @ ( F @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( F @ c_Groups_Oplus__class_Oplus ) ) @ ( F @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(271,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( A @ ( C @ c_Polynomial_Odegree ) @ c_Nat_OSuc @ ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__pCons__le) ).

thf(1882,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( A @ ( C @ c_Polynomial_Odegree ) @ c_Nat_OSuc @ ( A @ ( B @ ( C @ c_Polynomial_OpCons ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(550,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(2776,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)],[550]) ).

thf(642,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(3153,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)],[642]) ).

thf(844,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(3912,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)],[844]) ).

thf(519,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(2671,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)],[519]) ).

thf(220,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(1728,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)],[220]) ).

thf(1003,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(4479,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)],[1003]) ).

thf(156,axiom,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__add) ).

thf(1486,plain,
    tc_Int_Oint @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(776,axiom,
    tc_Nat_Onat @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ozero__neq__one) ).

thf(3680,plain,
    tc_Nat_Onat @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(872,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(4043,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)],[872]) ).

thf(741,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(3534,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)],[741]) ).

thf(777,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Oab__semigroup__mult ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__mult) ).

thf(3681,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Oab__semigroup__mult ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(861,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(3990,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)],[861]) ).

thf(188,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(1596,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)],[188]) ).

thf(341,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 ) ) )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__Suc__le__self) ).

thf(2104,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 ) ) )
       => ( ( C @ c_Groups_Oone__class_Oone @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ c_Nat_OSuc @ ( B @ ( C @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(950,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(4326,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)],[950]) ).

thf(734,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(3498,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)],[734]) ).

thf(302,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( 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__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__mono1) ).

thf(1971,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( 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__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(241,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(1793,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)],[241]) ).

thf(172,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__imp__le__nat) ).

thf(1533,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(922,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(4234,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)],[922]) ).

thf(719,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(3429,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)],[719]) ).

thf(524,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ! [D: $i] :
            ( ( D @ ( B @ hAPP ) @ hBOOL )
           => ( C @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ hAPP ) @ hBOOL ) )
       => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ! [D: $i] :
              ( ( D @ ( B @ hAPP ) @ hBOOL )
             => ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ hAPP ) @ hBOOL ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_incr__mult__lemma) ).

thf(2686,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( ! [D: $i] :
            ( ( D @ ( B @ hAPP ) @ hBOOL )
           => ( C @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ hAPP ) @ hBOOL ) )
       => ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ! [D: $i] :
              ( ( D @ ( B @ hAPP ) @ hBOOL )
             => ( C @ ( A @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( D @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ hAPP ) @ hBOOL ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(888,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(4110,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)],[888]) ).

thf(310,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(1999,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)],[310]) ).

thf(758,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(3616,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)],[758]) ).

thf(111,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less__mono) ).

thf(1342,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(686,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(3313,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)],[686]) ).

thf(23,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(1063,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)],[23]) ).

thf(116,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(1354,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)],[116]) ).

thf(146,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(1458,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)],[146]) ).

thf(237,axiom,
    tc_Int_Oint @ class_Groups_Ominus,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ominus) ).

thf(1788,plain,
    tc_Int_Oint @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(138,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(1426,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)],[138]) ).

thf(880,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(4090,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)],[880]) ).

thf(206,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__less__eq__trans) ).

thf(1655,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(866,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(4026,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)],[866]) ).

thf(616,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(3044,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)],[616]) ).

thf(505,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring) ).

thf(2627,plain,
    tc_Int_Oint @ class_Rings_Ocomm__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(362,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(2168,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)],[362]) ).

thf(71,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 ) ) )
        & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( 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_neg__mod__conj) ).

thf(1222,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 ) ) )
        & ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( A @ ( 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)],[71]) ).

thf(787,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(3717,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)],[787]) ).

thf(424,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__imp__le) ).

thf(2360,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(34,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(1095,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)],[34]) ).

thf(324,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(2046,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)],[324]) ).

thf(735,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(3502,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)],[735]) ).

thf(129,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1__strict,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1__strict) ).

thf(1391,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(806,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(3777,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)],[806]) ).

thf(680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( B @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__degree) ).

thf(3281,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
         != ( C @ c_Groups_Ozero__class_Ozero ) )
       => ( B @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__smult__le) ).

thf(3140,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__0 )
     => ( A @ ( C @ c_Polynomial_Odegree ) @ ( A @ ( B @ ( C @ c_Polynomial_Osmult ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(563,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(2825,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)],[563]) ).

thf(195,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(1623,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)],[195]) ).

thf(273,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(1887,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)],[273]) ).

thf(972,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(4386,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Odvd ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(825,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(3836,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)],[825]) ).

thf(957,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(4343,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)],[957]) ).

thf(419,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(2348,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)],[419]) ).

thf(538,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(2723,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)],[538]) ).

thf(79,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(1243,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)],[79]) ).

thf(45,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(1127,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)],[45]) ).

thf(546,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(2763,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)],[546]) ).

thf(845,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(3915,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)],[845]) ).

thf(877,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(4079,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)],[877]) ).

thf(933,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(4266,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)],[933]) ).

thf(289,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(1935,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)],[289]) ).

thf(854,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring) ).

thf(3947,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(899,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(4156,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)],[899]) ).

thf(245,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(1803,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)],[245]) ).

thf(836,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(3880,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)],[836]) ).

thf(280,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(1904,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)],[280]) ).

thf(168,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(1518,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)],[168]) ).

thf(85,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(1265,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(215,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(1703,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)],[215]) ).

thf(475,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(2531,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)],[475]) ).

thf(998,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(4467,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)],[998]) ).

thf(217,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(1709,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)],[217]) ).

thf(780,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(3689,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)],[780]) ).

thf(696,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(3342,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)],[696]) ).

thf(733,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(3495,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)],[733]) ).

thf(465,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(2501,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)],[465]) ).

thf(953,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(4334,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(634,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(3125,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)],[634]) ).

thf(795,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__imp__degree__le) ).

thf(3744,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( A
           != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
         => ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(350,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(2126,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)],[350]) ).

thf(830,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(3855,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)],[830]) ).

thf(924,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(4238,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)],[924]) ).

thf(21,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(1055,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)],[21]) ).

thf(133,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(1409,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)],[133]) ).

thf(674,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(3263,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)],[674]) ).

thf(988,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(4439,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)],[988]) ).

thf(500,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(2603,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)],[500]) ).

thf(585,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(2916,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)],[585]) ).

thf(718,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(3425,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)],[718]) ).

thf(240,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__idom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__idom) ).

thf(1792,plain,
    tc_Int_Oint @ class_Rings_Olinordered__idom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(140,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(1430,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)],[140]) ).

thf(528,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__le__mono) ).

thf(2698,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(96,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(1298,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)],[96]) ).

thf(531,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(2708,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)],[531]) ).

thf(820,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(3821,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)],[820]) ).

thf(105,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__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__le__divisor) ).

thf(1323,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__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(757,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(3612,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)],[757]) ).

thf(937,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(4275,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)],[937]) ).

thf(958,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(4349,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)],[958]) ).

thf(369,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_Groups_Oplus__class_Oplus ) ) @ ( 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_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) ).

thf(2185,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_Groups_Oplus__class_Oplus ) ) @ ( 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)],[369]) ).

thf(184,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = A )
      <=> ( B
          = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__imp__diff__is__add) ).

thf(1577,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
         => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
            = A ) )
        & ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
            = A )
         => ( B
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(145,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(1455,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)],[145]) ).

thf(487,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__square) ).

thf(2562,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(460,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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_Rings_Odvd__class_Odvd ) ) )
     => ( ( C @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__dvd__imp__le) ).

thf(2488,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( 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_Rings_Odvd__class_Odvd ) ) )
     => ( ( C @ ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(926,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(4243,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ocomm__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(530,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( B @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ hAPP ) @ hBOOL )
            & ! [D: $i] :
                ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
               => ~ ( D @ ( B @ hAPP ) @ hBOOL ) )
            & ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ex__least__nat__less) ).

thf(2703,plain,
    ! [A: $i,B: $i] :
      ( ~ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( B @ hAPP ) @ hBOOL )
     => ( ( A @ ( B @ hAPP ) @ hBOOL )
       => ? [C: $i] :
            ( ( tc_Nat_Onat @ c_Groups_Oone__class_Oone @ ( C @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ hAPP ) @ hBOOL )
            & ! [D: $i] :
                ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
               => ~ ( D @ ( B @ hAPP ) @ hBOOL ) )
            & ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(286,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(1922,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)],[286]) ).

thf(630,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(3099,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)],[630]) ).

thf(166,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(1514,plain,
    tc_Int_Oint @ class_Groups_Olinordered__ab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(325,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(2049,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)],[325]) ).

thf(154,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(1479,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)],[154]) ).

thf(428,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(2373,plain,
    ! [A: $i,B: $i] :
      ( ( ( B @ c_Nat_OSuc )
        = ( A @ c_Nat_OSuc ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(589,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__eq ) ) )
       => ( ( ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
            | ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__zero__or__degree__less) ).

thf(2929,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__eq ) ) )
       => ( ( ( A @ ( B @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
            = ( C @ c_Groups_Ozero__class_Ozero ) )
         => ( ( A @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
            | ( B
              = ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(770,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(3657,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)],[770]) ).

thf(234,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(1773,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)],[234]) ).

thf(974,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(4390,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)],[974]) ).

thf(570,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(2852,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)],[570]) ).

thf(832,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(3866,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)],[832]) ).

thf(260,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__eq__less__or__eq) ).

thf(1847,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(533,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( 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_q__pos__lemma) ).

thf(2713,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ ( 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)],[533]) ).

thf(890,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Oplus ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Oplus) ).

thf(4119,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Oplus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(124,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1) ).

thf(1379,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(709,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(3385,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)],[709]) ).

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(1013,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(935,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(4270,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)],[935]) ).

thf(225,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_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( 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_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) ).

thf(1741,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_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( 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)],[225]) ).

thf(626,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(3084,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)],[626]) ).

thf(791,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(3734,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)],[791]) ).

thf(964,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(4368,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)],[964]) ).

thf(610,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(3014,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)],[610]) ).

thf(74,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(1232,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)],[74]) ).

thf(930,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(4253,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)],[930]) ).

thf(672,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(3258,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)],[672]) ).

thf(690,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(3322,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)],[690]) ).

thf(555,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(2794,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(604,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(2996,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)],[604]) ).

thf(49,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)],[49]) ).

thf(253,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(1828,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)],[253]) ).

thf(52,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I3_J) ).

thf(1151,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(397,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(2291,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)],[397]) ).

thf(381,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(2222,plain,
    tc_Int_Oint @ class_Rings_Oring__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(980,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(4411,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)],[980]) ).

thf(951,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(4329,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)],[951]) ).

thf(409,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__power__le) ).

thf(2324,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Ocomm__semiring__1 )
     => ( A @ ( B @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ tc_Polynomial_Opoly @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(249,axiom,
    tc_Int_Oint @ class_Groups_Ogroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ogroup__add) ).

thf(1811,plain,
    tc_Int_Oint @ class_Groups_Ogroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(143,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(1450,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)],[143]) ).

thf(578,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(2893,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)],[578]) ).

thf(943,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(4297,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)],[943]) ).

thf(450,axiom,
    tc_Int_Oint @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__mult) ).

thf(2451,plain,
    tc_Int_Oint @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(384,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(2236,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)],[384]) ).

thf(721,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(3438,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)],[721]) ).

thf(739,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(3525,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)],[739]) ).

thf(348,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__mono1) ).

thf(2122,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(682,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(3290,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)],[682]) ).

thf(629,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(3093,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)],[629]) ).

thf(93,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(1283,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)],[93]) ).

thf(333,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(2073,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)],[333]) ).

thf(19,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(1051,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)],[19]) ).

thf(901,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(4162,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)],[901]) ).

thf(742,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(3537,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)],[742]) ).

thf(920,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(4230,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(69,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(1215,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(277,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(1896,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)],[277]) ).

thf(458,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(2483,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)],[458]) ).

thf(736,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(3505,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)],[736]) ).

thf(42,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(1119,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)],[42]) ).

thf(548,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(2772,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) )
        = ( B @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(115,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__diff__cancel) ).

thf(1351,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(395,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add2) ).

thf(2286,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(692,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(3327,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)],[692]) ).

thf(545,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(2750,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)],[545]) ).

thf(78,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(1242,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(536,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(2718,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)],[536]) ).

thf(1001,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(4475,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)],[1001]) ).

thf(878,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(4082,plain,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(708,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(3378,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)],[708]) ).

thf(816,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(3808,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)],[816]) ).

thf(242,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(1796,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)],[242]) ).

thf(25,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(1072,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)],[25]) ).

thf(871,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(4040,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)],[871]) ).

thf(800,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Omonoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Omonoid__add) ).

thf(3756,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(663,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(3226,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)],[663]) ).

thf(292,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__le__mono2) ).

thf(1943,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(66,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(1201,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)],[66]) ).

thf(754,axiom,
    tc_Nat_Onat @ class_Groups_Oplus,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oplus) ).

thf(3606,plain,
    tc_Nat_Onat @ class_Groups_Oplus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(660,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(3211,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)],[660]) ).

thf(432,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(2392,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)],[432]) ).

thf(720,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(3435,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)],[720]) ).

thf(598,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(2954,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)],[598]) ).

thf(664,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(3229,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)],[664]) ).

thf(913,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(4199,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(583,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(2910,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)],[583]) ).

thf(375,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(2204,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)],[375]) ).

thf(457,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( B
          = ( A @ c_Nat_OSuc ) )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__Suc__eq) ).

thf(2473,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B
            = ( A @ c_Nat_OSuc ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B
            = ( A @ c_Nat_OSuc ) )
          | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(213,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(1691,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)],[213]) ).

thf(788,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(3720,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)],[788]) ).

thf(648,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(3169,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)],[648]) ).

thf(425,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(2364,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)],[425]) ).

thf(53,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__imp__less__Suc) ).

thf(1153,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(151,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(1470,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)],[151]) ).

thf(511,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__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__Suc__eq__le) ).

thf(2641,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( 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 ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(26,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(1074,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)],[26]) ).

thf(191,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add1__zle__eq) ).

thf(1604,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(829,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(3852,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)],[829]) ).

thf(97,axiom,
    tc_Int_Oint @ class_Rings_Oordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__semiring) ).

thf(1306,plain,
    tc_Int_Oint @ class_Rings_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(235,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__le__dvd) ).

thf(1783,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( D @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( D @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(965,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(4370,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)],[965]) ).

thf(805,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(3767,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)],[805]) ).

thf(403,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(2307,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)],[403]) ).

thf(336,axiom,
    tc_Int_Oint @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring__0) ).

thf(2083,plain,
    tc_Int_Oint @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(868,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ouminus )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Groups_Ouminus__class_Ouminus ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( D @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_uminus__apply) ).

thf(4031,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Groups_Ouminus )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Groups_Ouminus__class_Ouminus ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( D @ c_Groups_Ouminus__class_Ouminus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(223,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__eq ) ) )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__leD1) ).

thf(1736,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__eq ) ) )
     => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(463,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__one__le__power) ).

thf(2496,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(553,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__cancel__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__cancel__semiring) ).

thf(2790,plain,
    tc_Nat_Onat @ class_Rings_Oordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(581,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_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Polynomial_Opcompose ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__pcompose__le) ).

thf(2905,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_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( B @ ( C @ c_Polynomial_Opcompose ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(264,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(1868,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)],[264]) ).

thf(329,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(2061,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)],[329]) ).

thf(199,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(1636,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)],[199]) ).

thf(17,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(1047,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)],[17]) ).

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

thf(3679,plain,
    tc_Nat_Onat @ class_Rings_Odvd,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(15,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( 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_power__less__imp__less__base) ).

thf(1042,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( 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)],[15]) ).

thf(986,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(4434,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)],[986]) ).

thf(356,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(2147,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)],[356]) ).

thf(613,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(3022,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)],[613]) ).

thf(157,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(1487,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)],[157]) ).

thf(705,axiom,
    tc_Nat_Onat @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__add) ).

thf(3371,plain,
    tc_Nat_Onat @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(818,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ? [D: $i] :
            ( ( ( D @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
             != ( C @ c_Groups_Ozero__class_Ozero ) )
            & ( D @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__degree__imp) ).

thf(3814,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ( A @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ? [D: $i] :
            ( ( ( D @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
             != ( C @ c_Groups_Ozero__class_Ozero ) )
            & ( D @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(729,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(3471,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)],[729]) ).

thf(446,axiom,
    tc_Int_Oint @ class_Groups_Oplus,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oplus) ).

thf(2441,plain,
    tc_Int_Oint @ class_Groups_Oplus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(520,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__le__eq) ).

thf(2673,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(822,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(3827,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)],[822]) ).

thf(443,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(2428,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)],[443]) ).

thf(785,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(3711,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)],[785]) ).

thf(486,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(2560,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)],[486]) ).

thf(559,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(2805,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)],[559]) ).

thf(210,axiom,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__eq__nat_Osimps_I1_J) ).

thf(1676,plain,
    ! [A: $i] : ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(351,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__diff) ).

thf(2133,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(992,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(4449,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)],[992]) ).

thf(635,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__degree) ).

thf(3134,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Oidom )
     => ( ( B
         != ( C @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
       => ( B @ ( C @ c_Polynomial_Odegree ) @ ( B @ ( A @ ( C @ c_Polynomial_Oorder ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(940,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(4284,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)],[940]) ).

thf(194,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(1619,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)],[194]) ).

thf(357,axiom,
    tc_Int_Oint @ class_Groups_Oone,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oone) ).

thf(2150,plain,
    tc_Int_Oint @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(439,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(2414,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)],[439]) ).

thf(670,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(3250,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)],[670]) ).

thf(695,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(3339,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)],[695]) ).

thf(317,axiom,
    tc_Int_Oint @ class_Groups_Omonoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__add) ).

thf(2015,plain,
    tc_Int_Oint @ class_Groups_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(506,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(2628,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)],[506]) ).

thf(873,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(4053,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)],[873]) ).

thf(628,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(3090,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)],[628]) ).

thf(179,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
      | ~ ( A @ ( B @ c_fequal ) @ hBOOL ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__fequal__1) ).

thf(1555,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
      | ~ ( A @ ( B @ c_fequal ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(574,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(2872,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)],[574]) ).

thf(331,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(2068,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)],[331]) ).

thf(639,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(3144,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)],[639]) ).

thf(367,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 @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__power__le) ).

thf(2181,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 @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( E @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( E @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(8,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semidom) ).

thf(1027,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(130,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(1392,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)],[130]) ).

thf(38,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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_inverse__nonpositive__iff__nonpositive) ).

thf(1106,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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)],[38]) ).

thf(855,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(3948,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(685,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(3307,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)],[685]) ).

thf(677,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(3271,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)],[677]) ).

thf(647,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(3166,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)],[647]) ).

thf(448,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__diffD) ).

thf(2445,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(942,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(4295,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)],[942]) ).

thf(654,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(3191,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)],[654]) ).

thf(454,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(2464,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)],[454]) ).

thf(290,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(1938,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)],[290]) ).

thf(10,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(1032,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)],[10]) ).

thf(255,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(1837,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)],[255]) ).

thf(676,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(3269,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)],[676]) ).

thf(636,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(3137,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)],[636]) ).

thf(797,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(3750,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)],[797]) ).

thf(697,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(3349,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)],[697]) ).

thf(745,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(3556,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)],[745]) ).

thf(749,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(3578,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)],[749]) ).

thf(60,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_fequal ) @ hBOOL )
      | ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__fequal__2) ).

thf(1178,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_fequal ) @ hBOOL )
      | ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(982,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(4417,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)],[982]) ).

thf(618,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(3050,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)],[618]) ).

thf(377,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(2211,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)],[377]) ).

thf(132,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oord )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oord ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Orderings_Oord) ).

thf(1407,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Orderings_Oord )
     => ( B @ ( A @ tc_fun ) @ class_Orderings_Oord ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(479,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(2547,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)],[479]) ).

thf(406,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(2313,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)],[406]) ).

thf(885,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(4104,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)],[885]) ).

thf(92,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__le__lessD) ).

thf(1281,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(542,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(2741,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)],[542]) ).

thf(614,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(3033,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)],[614]) ).

thf(714,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(3408,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)],[714]) ).

thf(282,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(1910,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)],[282]) ).

thf(547,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(2769,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)],[547]) ).

thf(949,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(4316,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)],[949]) ).

thf(790,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(3731,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)],[790]) ).

thf(892,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Oab__semigroup__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__add) ).

thf(4123,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Oab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(566,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(2840,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)],[566]) ).

thf(231,axiom,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__neg__neg__trivial) ).

thf(1762,plain,
    ! [A: $i,B: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( A @ ( B @ ( tc_Int_Oint @ c_Divides_Odiv__class_Omod ) ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(928,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(4248,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)],[928]) ).

thf(1006,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(4486,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)],[1006]) ).

thf(301,axiom,
    tc_Nat_Onat @ class_Divides_Osemiring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Divides_Osemiring__div) ).

thf(1970,plain,
    tc_Nat_Onat @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(631,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(3102,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)],[631]) ).

thf(507,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(2631,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)],[507]) ).

thf(624,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(3078,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)],[624]) ).

thf(767,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(3650,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)],[767]) ).

thf(510,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(2638,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)],[510]) ).

thf(783,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(3699,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)],[783]) ).

thf(400,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( 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 ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__imp__le) ).

thf(2300,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( 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 ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(843,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(3909,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)],[843]) ).

thf(808,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(3782,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)],[808]) ).

thf(32,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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_inverse__negative__iff__negative) ).

thf(1089,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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)],[32]) ).

thf(464,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(2498,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)],[464]) ).

thf(87,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)],[87]) ).

thf(633,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(3122,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)],[633]) ).

thf(6,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__le__inverse__iff) ).

thf(1017,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
         => ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(296,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(1953,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)],[296]) ).

thf(526,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(2692,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)],[526]) ).

thf(161,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(1498,plain,
    tc_Int_Oint @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(175,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(1540,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)],[175]) ).

thf(738,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(3518,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)],[738]) ).

thf(882,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(4095,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)],[882]) ).

thf(632,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(3119,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)],[632]) ).

thf(658,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(3205,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)],[658]) ).

thf(915,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(4219,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)],[915]) ).

thf(651,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(3176,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)],[651]) ).

thf(627,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(3087,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)],[627]) ).

thf(183,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( ( B != A )
        & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__less__le) ).

thf(1567,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( B != A )
          & ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(107,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(1329,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)],[107]) ).

thf(576,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(2877,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)],[576]) ).

thf(119,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(1361,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)],[119]) ).

thf(652,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(3179,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)],[652]) ).

thf(827,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(3846,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)],[827]) ).

thf(713,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__comm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__comm__semiring) ).

thf(3407,plain,
    tc_Nat_Onat @ class_Rings_Oordered__comm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(61,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(1182,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)],[61]) ).

thf(147,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat_Odiff__diff__eq) ).

thf(1461,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(786,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(3714,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)],[786]) ).

thf(335,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(2080,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)],[335]) ).

thf(258,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(1842,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)],[258]) ).

thf(978,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(4406,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)],[978]) ).

thf(239,axiom,
    tc_Int_Oint @ class_Groups_Oab__group__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__group__add) ).

thf(1791,plain,
    tc_Int_Oint @ class_Groups_Oab__group__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(655,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(3194,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)],[655]) ).

thf(344,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(2108,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)],[344]) ).

thf(274,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(1889,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)],[274]) ).

thf(54,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__mono2) ).

thf(1155,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(452,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( 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_zless__imp__add1__zle) ).

thf(2460,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( B @ ( 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)],[452]) ).

thf(159,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(1493,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)],[159]) ).

thf(558,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(2798,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)],[558]) ).

thf(9,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(1028,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)],[9]) ).

thf(945,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(4301,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)],[945]) ).

thf(319,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(2023,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)],[319]) ).

thf(300,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(1964,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)],[300]) ).

thf(207,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(1659,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)],[207]) ).

thf(847,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(3922,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)],[847]) ).

thf(722,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(3441,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)],[722]) ).

thf(609,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(3010,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)],[609]) ).

thf(482,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(2551,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)],[482]) ).

thf(68,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(1207,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)],[68]) ).

thf(789,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(3723,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)],[789]) ).

thf(766,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(3643,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)],[766]) ).

thf(990,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(4444,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)],[990]) ).

thf(517,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(2666,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)],[517]) ).

thf(193,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(1612,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)],[193]) ).

thf(908,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(4180,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)],[908]) ).

thf(14,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(1039,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)],[14]) ).

thf(466,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(2504,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)],[466]) ).

thf(84,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(1263,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)],[84]) ).

thf(971,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(4384,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__ring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(522,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__diff__le) ).

thf(2681,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ c_Nat_OSuc ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(402,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(2304,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)],[402]) ).

thf(47,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(1131,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)],[47]) ).

thf(178,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(1552,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)],[178]) ).

thf(819,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(3819,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)],[819]) ).

thf(243,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(1799,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)],[243]) ).

thf(294,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(1947,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)],[294]) ).

thf(306,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(1984,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)],[306]) ).

thf(189,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(1599,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(621,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(3064,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)],[621]) ).

thf(607,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(3005,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)],[607]) ).

thf(354,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( 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__is__0__eq_H) ).

thf(2140,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(427,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(2370,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)],[427]) ).

thf(590,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(2932,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)],[590]) ).

thf(731,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(3486,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)],[731]) ).

thf(769,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(3654,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)],[769]) ).

thf(186,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(1588,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)],[186]) ).

thf(36,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(1099,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)],[36]) ).

thf(565,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(2836,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)],[565]) ).

thf(126,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(1382,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)],[126]) ).

thf(699,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(3354,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)],[699]) ).

thf(359,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(2153,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)],[359]) ).

thf(579,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(2900,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)],[579]) ).

thf(422,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I4_J) ).

thf(2355,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(64,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(1192,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)],[64]) ).

thf(662,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(3222,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)],[662]) ).

thf(314,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(2006,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)],[314]) ).

thf(669,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(3247,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)],[669]) ).

thf(572,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(2862,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)],[572]) ).

thf(656,axiom,
    tc_Nat_Onat @ class_Rings_Omult__zero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Omult__zero) ).

thf(3201,plain,
    tc_Nat_Onat @ class_Rings_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(917,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(4223,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)],[917]) ).

thf(230,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(1759,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)],[230]) ).

thf(640,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(3147,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)],[640]) ).

thf(109,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(1333,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)],[109]) ).

thf(103,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(1318,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)],[103]) ).

thf(588,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(2926,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)],[588]) ).

thf(959,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(4352,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(762,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(3624,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)],[762]) ).

thf(577,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(2883,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)],[577]) ).

thf(182,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(1563,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(327,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(2057,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)],[327]) ).

thf(334,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(2076,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)],[334]) ).

thf(149,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(1466,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)],[149]) ).

thf(437,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(2408,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)],[437]) ).

thf(167,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(1515,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)],[167]) ).

thf(717,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(3422,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)],[717]) ).

thf(309,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__diff__inverse) ).

thf(1996,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(815,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(3807,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(821,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(3824,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)],[821]) ).

thf(842,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(3903,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)],[842]) ).

thf(261,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(1857,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)],[261]) ).

thf(94,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(1286,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)],[94]) ).

thf(689,axiom,
    tc_Nat_Onat @ class_Rings_Osemiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring__0) ).

thf(3321,plain,
    tc_Nat_Onat @ class_Rings_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

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

thf(2368,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(516,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(2662,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)],[516]) ).

thf(62,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(1185,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)],[62]) ).

thf(441,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(2423,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)],[441]) ).

thf(474,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(2529,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)],[474]) ).

thf(389,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(2253,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)],[389]) ).

thf(838,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(3890,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)],[838]) ).

thf(89,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(1273,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)],[89]) ).

thf(490,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(2570,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)],[490]) ).

thf(288,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(1927,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)],[288]) ).

thf(779,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(3686,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)],[779]) ).

thf(122,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(1374,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)],[122]) ).

thf(91,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(1278,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)],[91]) ).

thf(281,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__fun__def) ).

thf(1906,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ! [E: $i] : ( E @ ( A @ hAPP ) @ ( E @ ( B @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(592,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(2936,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)],[592]) ).

thf(539,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(2729,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)],[539]) ).

thf(771,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(3660,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)],[771]) ).

thf(311,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(2001,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)],[311]) ).

thf(999,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(4470,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)],[999]) ).

thf(229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Olinordered__field__inverse__zero )
     => ( ! [D: $i] :
            ( ( D @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
           => ( ( C @ c_Groups_Oone__class_Oone @ ( D @ ( C @ c_Orderings_Oord__class_Oless ) ) )
             => ( A @ ( B @ ( D @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_field__le__mult__one__interval) ).

thf(1754,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Fields_Olinordered__field__inverse__zero )
     => ( ! [D: $i] :
            ( ( D @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless ) ) )
           => ( ( C @ c_Groups_Oone__class_Oone @ ( D @ ( C @ c_Orderings_Oord__class_Oless ) ) )
             => ( A @ ( B @ ( D @ ( C @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) )
       => ( A @ ( B @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(969,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(4380,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oordered__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(484,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(2556,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)],[484]) ).

thf(960,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(4354,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)],[960]) ).

thf(671,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__1) ).

thf(3257,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(70,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(1219,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)],[70]) ).

thf(755,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(3607,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)],[755]) ).

thf(956,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(4340,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)],[956]) ).

thf(707,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(3375,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)],[707]) ).

thf(826,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(3839,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)],[826]) ).

thf(962,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(4359,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)],[962]) ).

thf(678,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(3274,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)],[678]) ).

thf(471,axiom,
    tc_Int_Oint @ class_Rings_Olinordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring) ).

thf(2524,plain,
    tc_Int_Oint @ class_Rings_Olinordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(994,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(4453,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)],[994]) ).

thf(706,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(3372,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)],[706]) ).

thf(863,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(4000,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)],[863]) ).

thf(794,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(3741,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)],[794]) ).

thf(688,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Ozero) ).

thf(3319,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(284,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(1916,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)],[284]) ).

thf(272,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 @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_self__quotient__aux2) ).

thf(1884,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 @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(44,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(1125,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)],[44]) ).

thf(228,axiom,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add1) ).

thf(1752,plain,
    ! [A: $i,B: $i] : ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(318,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
        & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__le__mult__iff) ).

thf(2016,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(128,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(1388,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)],[128]) ).

thf(173,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(1535,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)],[173]) ).

thf(727,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(3458,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)],[727]) ).

thf(535,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) ).

thf(2716,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Int_Oint @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(712,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(3404,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)],[712]) ).

thf(934,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(4268,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)],[934]) ).

thf(623,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(3070,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)],[623]) ).

thf(817,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(3811,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)],[817]) ).

thf(554,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(2791,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)],[554]) ).

thf(472,axiom,
    tc_Int_Oint @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__semigroup__add) ).

thf(2525,plain,
    tc_Int_Oint @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(756,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(3609,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)],[756]) ).

thf(900,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(4159,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)],[900]) ).

thf(569,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(2849,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)],[569]) ).

thf(205,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__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__leD2) ).

thf(1652,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__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(625,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(3081,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)],[625]) ).

thf(801,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(3758,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)],[801]) ).

thf(997,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(4465,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)],[997]) ).

thf(841,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(3900,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)],[841]) ).

thf(527,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(2695,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)],[527]) ).

thf(72,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(1226,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)],[72]) ).

thf(410,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(2326,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)],[410]) ).

thf(100,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(1312,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)],[100]) ).

thf(22,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless ) ) )
          | ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_inverse__less__1__iff) ).

thf(1058,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless ) ) )
            | ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless ) ) )
            | ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(750,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(3594,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)],[750]) ).

thf(447,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(2442,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)],[447]) ).

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(4060,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(684,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(3304,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)],[684]) ).

thf(148,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__funE) ).

thf(1464,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(117,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I5_J) ).

thf(1356,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(740,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__eq ) ) )
       => ( ( B @ ( A @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__add__le) ).

thf(3532,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__eq ) ) )
       => ( ( B @ ( A @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( A @ ( C @ ( D @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) @ ( D @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(139,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(1428,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)],[139]) ).

thf(929,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(4251,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)],[929]) ).

thf(50,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)],[50]) ).

thf(811,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Ocomm__semiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Rings_Ocomm__semiring) ).

thf(3796,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Ocomm__semiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(163,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(1503,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)],[163]) ).

thf(905,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(4171,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)],[905]) ).

thf(659,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(3208,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)],[659]) ).

thf(923,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Oord ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Oord) ).

thf(4236,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Orderings_Oord ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(772,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(3663,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)],[772]) ).

thf(263,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(1865,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)],[263]) ).

thf(591,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(2935,plain,
    tc_Nat_Onat @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(606,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(3002,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)],[606]) ).

thf(222,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(1733,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)],[222]) ).

thf(232,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(1765,plain,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Osemiring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(759,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(3619,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(936,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(4273,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__ring )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(82,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__assoc) ).

thf(1258,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(200,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__eq ) ) )
      <=> ( 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__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__iff2) ).

thf(1638,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__eq ) ) )
         => ( 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__eq ) ) ) )
        & ( ( 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__eq ) ) )
         => ( 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__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(927,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(4245,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)],[927]) ).

thf(468,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(2510,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)],[468]) ).

thf(405,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(2311,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)],[405]) ).

thf(470,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(2521,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)],[470]) ).

thf(521,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(2679,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)],[521]) ).

thf(919,axiom,
    ( ( v_h @ ( v_p @ ( t_a @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( v_a @ ( t_a @ c_Polynomial_OpCons ) ) @ ( v_h @ ( v_p @ ( t_a @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( v_h @ ( t_a @ c_Polynomial_Osmult ) ) @ ( t_a @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
    = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).

thf(4228,plain,
    ( ( v_h @ ( v_p @ ( t_a @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( v_a @ ( t_a @ c_Polynomial_OpCons ) ) @ ( v_h @ ( v_p @ ( t_a @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) @ ( v_h @ ( t_a @ c_Polynomial_Osmult ) ) @ ( t_a @ tc_Polynomial_Opoly @ c_Groups_Oplus__class_Oplus ) ) )
    = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(358,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( 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__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__mono) ).

thf(2151,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( C @ ( D @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( 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__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(337,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(2084,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)],[337]) ).

thf(41,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(1117,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)],[41]) ).

thf(716,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(3419,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)],[716]) ).

thf(114,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(1349,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)],[114]) ).

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

thf(1603,plain,
    tc_Int_Oint @ class_Rings_Oordered__comm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(345,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(2116,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)],[345]) ).

thf(595,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(2945,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)],[595]) ).

thf(43,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( C = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__inject__base) ).

thf(1122,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( C = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(973,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(4388,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)],[973]) ).

thf(65,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(1194,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)],[65]) ).

thf(417,axiom,
    tc_Int_Oint @ class_Divides_Oring__div,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Divides_Oring__div) ).

thf(2346,plain,
    tc_Int_Oint @ class_Divides_Oring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(346,axiom,
    tc_Int_Oint @ class_Rings_Oring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1) ).

thf(2119,plain,
    tc_Int_Oint @ class_Rings_Oring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(698,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(3352,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)],[698]) ).

thf(803,axiom,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__mult) ).

thf(3764,plain,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(462,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(2493,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)],[462]) ).

thf(246,axiom,
    tc_Int_Oint @ class_Rings_Odvd,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Odvd) ).

thf(1806,plain,
    tc_Int_Oint @ class_Rings_Odvd,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

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(1011,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(858,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(3957,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)],[858]) ).

thf(778,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(3683,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)],[778]) ).

thf(169,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__eq ) ) )
      <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__le__cancel1) ).

thf(1524,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__eq ) ) )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(518,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(2669,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)],[518]) ).

thf(896,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(4139,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)],[896]) ).

thf(51,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(1144,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)],[51]) ).

thf(244,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_Omod ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__mod__sign) ).

thf(1801,plain,
    ! [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_Omod ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(898,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Omult__zero ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Rings_Omult__zero) ).

thf(4154,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Omult__zero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(429,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( 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__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__cancel1) ).

thf(2378,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__eq ) ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( 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__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(967,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(4374,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)],[967]) ).

thf(622,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(3066,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)],[622]) ).

thf(123,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(1376,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)],[123]) ).

thf(142,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(1439,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)],[142]) ).

thf(887,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(4107,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)],[887]) ).

thf(98,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(1307,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)],[98]) ).

thf(372,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(2197,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)],[372]) ).

thf(1002,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(4477,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)],[1002]) ).

thf(366,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(2180,plain,
    tc_Int_Oint @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(661,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(3219,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)],[661]) ).

thf(796,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(3747,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)],[796]) ).

thf(391,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
     => ( ( A @ ( B @ ( 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 @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( D @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
               => ( B @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__mono2__lemma) ).

thf(2267,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
     => ( ( A @ ( B @ ( 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 @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( D @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
               => ( B @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(952,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(4332,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(106,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__eq ) ) )
      <=> ( 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__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__iff1) ).

thf(1325,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__eq ) ) )
         => ( 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__eq ) ) ) )
        & ( ( 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__eq ) ) )
         => ( 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__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(724,axiom,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__mult) ).

thf(3451,plain,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(112,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(1344,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)],[112]) ).

thf(7,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( 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_one__less__inverse__iff) ).

thf(1022,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Orderings_Oord__class_Oless ) ) )
            & ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
         => ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( 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)],[7]) ).

thf(393,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ 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_Suc__leD) ).

thf(2278,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ 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)],[393]) ).

thf(601,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(2974,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)],[601]) ).

thf(983,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(4420,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)],[983]) ).

thf(501,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(2605,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)],[501]) ).

thf(416,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(2343,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)],[416]) ).

thf(774,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(3668,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)],[774]) ).

thf(504,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__trans) ).

thf(2625,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(221,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__funD) ).

thf(1731,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(932,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(4264,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)],[932]) ).

thf(683,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(3298,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)],[683]) ).

thf(575,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(2874,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)],[575]) ).

thf(477,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(2542,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)],[477]) ).

thf(512,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ? [C: $i] :
          ( A
          = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__Suc__ex__iff) ).

thf(2647,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(641,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(3150,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)],[641]) ).

thf(411,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(2329,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)],[411]) ).

thf(839,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(3893,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)],[839]) ).

thf(532,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(2710,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)],[532]) ).

thf(40,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( 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_inverse__le__1__iff) ).

thf(1112,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
            | ( B @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( B @ c_Groups_Oone__class_Oone @ ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( B @ c_Groups_Oone__class_Oone @ ( 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)],[40]) ).

thf(492,axiom,
    tc_Int_Oint @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Power_Opower) ).

thf(2578,plain,
    tc_Int_Oint @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(212,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(1688,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)],[212]) ).

thf(57,axiom,
    tc_Int_Oint @ class_Rings_Oordered__cancel__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__cancel__semiring) ).

thf(1171,plain,
    tc_Int_Oint @ class_Rings_Oordered__cancel__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(921,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(4232,plain,
    ! [A: $i] :
      ( ( A @ class_Fields_Ofield )
     => ( A @ tc_Polynomial_Opoly @ class_Divides_Oring__div ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(434,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( 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 @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( F @ ( D @ ( 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__eq ) ) )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
               => ( E @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__mono2__neg__lemma) ).

thf(2401,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( D @ ( E @ ( F @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) )
        = ( A @ ( B @ ( 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 @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( F @ ( D @ ( 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__eq ) ) )
           => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
             => ( ( F @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
               => ( E @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(564,axiom,
    tc_Nat_Onat @ class_Groups_Ominus,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ominus) ).

thf(2835,plain,
    tc_Nat_Onat @ class_Groups_Ominus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(170,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__diff__inverse2) ).

thf(1528,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(59,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(1176,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)],[59]) ).

thf(396,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__diff__assoc2) ).

thf(2288,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(638,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Osemiring ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Rings_Osemiring) ).

thf(3142,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Osemiring ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(476,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
            = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__diff__iff) ).

thf(2535,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( B = A )
           => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
              = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) )
          & ( ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
              = ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(120,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(1364,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)],[120]) ).

thf(160,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ouminus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ouminus ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Groups_Ouminus) ).

thf(1496,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ouminus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ouminus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(37,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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_inverse__nonnegative__iff__nonnegative) ).

thf(1102,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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)],[37]) ).

thf(360,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__eq ) ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__cancel__le) ).

thf(2156,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( 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__eq ) ) ) )
      & ( ( 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__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(376,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(2207,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(1000,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(4473,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)],[1000]) ).

thf(763,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(3627,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)],[763]) ).

thf(363,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(2171,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)],[363]) ).

thf(16,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(1044,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)],[16]) ).

thf(481,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(2550,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)],[481]) ).

thf(80,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B
          = ( A @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__SucE) ).

thf(1245,plain,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ~ ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B
          = ( A @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(966,axiom,
    t_a @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tfree_0) ).

thf(4373,plain,
    t_a @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(202,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(1643,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)],[202]) ).

thf(644,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(3159,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)],[644]) ).

thf(330,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__eq ) ) )
     => ~ ( ( A @ ( C @ ( tc_Nat_Onat @ 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_add__leE) ).

thf(2064,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__eq ) ) )
     => ~ ( ( A @ ( C @ ( tc_Nat_Onat @ 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)],[330]) ).

thf(5,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__le__imp__le__base) ).

thf(1015,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ c_Nat_OSuc @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( B @ c_Nat_OSuc @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(667,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(3241,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)],[667]) ).

thf(557,axiom,
    tc_Nat_Onat @ class_Orderings_Oord,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oord) ).

thf(2797,plain,
    tc_Nat_Onat @ class_Orderings_Oord,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(611,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(3016,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)],[611]) ).

thf(508,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(2633,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)],[508]) ).

thf(702,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(3363,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)],[702]) ).

thf(602,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(2984,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)],[602]) ).

thf(303,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__apply) ).

thf(1973,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E @ class_Groups_Ominus )
     => ( ( A @ ( B @ ( C @ ( E @ ( D @ tc_fun ) @ c_Groups_Ominus__class_Ominus ) ) @ hAPP ) )
        = ( A @ ( B @ hAPP ) @ ( A @ ( C @ hAPP ) @ ( E @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(1004,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(4481,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)],[1004]) ).

thf(985,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(4426,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)],[985]) ).

thf(248,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__cube) ).

thf(1809,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(266,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(1873,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)],[266]) ).

thf(321,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__le__eq__trans) ).

thf(2037,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B = A )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(349,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(2124,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)],[349]) ).

thf(726,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(3455,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)],[726]) ).

thf(954,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(4336,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)],[954]) ).

thf(313,axiom,
    tc_Int_Oint @ class_Rings_Ozero__neq__one,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ozero__neq__one) ).

thf(2005,plain,
    tc_Int_Oint @ class_Rings_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(136,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(1413,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)],[136]) ).

thf(196,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__diff__right) ).

thf(1626,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(608,axiom,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__add) ).

thf(3009,plain,
    tc_Nat_Onat @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(11,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(1034,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)],[11]) ).

thf(385,axiom,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
    <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__is__0__eq) ).

thf(2239,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) )
      & ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(208,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(1662,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)],[208]) ).

thf(895,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(4136,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)],[895]) ).

thf(852,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(3936,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)],[852]) ).

thf(343,axiom,
    tc_Int_Oint @ class_Int_Oring__char__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Int_Oring__char__0) ).

thf(2107,plain,
    tc_Int_Oint @ class_Int_Oring__char__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(408,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ 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_Suc__le__mono) ).

thf(2318,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ c_Nat_OSuc @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ 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)],[408]) ).

thf(995,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(4456,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)],[995]) ).

thf(645,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(3161,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)],[645]) ).

thf(137,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(1420,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)],[137]) ).

thf(28,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(1081,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)],[28]) ).

thf(907,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(4177,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)],[907]) ).

thf(308,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
    <=> ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__eq__Suc__le) ).

thf(1990,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(523,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(2684,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)],[523]) ).

thf(374,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( B @ ( 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__right__le__one__le) ).

thf(2202,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( B @ ( 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)],[374]) ).

thf(413,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
        <=> ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__diff__iff) ).

thf(2333,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
          & ( ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(568,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(2846,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)],[568]) ).

thf(889,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(4113,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)],[889]) ).

thf(488,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(2564,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)],[488]) ).

thf(498,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(2597,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)],[498]) ).

thf(298,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(1958,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)],[298]) ).

thf(904,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(4168,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( C @ ( B @ hAPP ) )
          = ( C @ ( A @ hAPP ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(494,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(2582,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)],[494]) ).

thf(747,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(3571,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)],[747]) ).

thf(828,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(3849,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)],[828]) ).

thf(18,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(1049,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)],[18]) ).

thf(594,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(2942,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)],[594]) ).

thf(710,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(3391,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)],[710]) ).

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__imp__eq) ).

thf(3737,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)],[792]) ).

thf(534,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(2715,plain,
    tc_Int_Oint @ class_Groups_Oordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(765,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(3640,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)],[765]) ).

thf(299,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(1961,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)],[299]) ).

thf(355,axiom,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__n__not__n) ).

thf(2143,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(203,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(1647,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)],[203]) ).

thf(824,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(3833,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)],[824]) ).

thf(364,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ominus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ominus ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Groups_Ominus) ).

thf(2175,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Groups_Ominus )
     => ( B @ ( A @ tc_fun ) @ class_Groups_Ominus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(171,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ 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_nat__le__linear) ).

thf(1531,plain,
    ! [A: $i,B: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ 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)],[171]) ).

thf(860,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(3988,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)],[860]) ).

thf(259,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__le__add1) ).

thf(1845,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(469,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(2513,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)],[469]) ).

thf(489,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(2567,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)],[489]) ).

thf(226,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(1743,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)],[226]) ).

thf(963,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(4366,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Oidom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Ono__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(673,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(3260,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)],[673]) ).

thf(799,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(3754,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)],[799]) ).

thf(436,axiom,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J) ).

thf(2407,plain,
    tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(430,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(2386,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)],[430]) ).

thf(537,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(2720,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)],[537]) ).

thf(834,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(3872,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)],[834]) ).

thf(968,axiom,
    ( ( ( v_h @ ( v_p @ ( t_a @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
      = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
   => ( v_p
      = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(4377,plain,
    ( ( ( v_h @ ( v_p @ ( t_a @ c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly ) ) )
      = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
   => ( v_p
      = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(723,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(3448,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)],[723]) ).

thf(912,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(4190,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)],[912]) ).

thf(693,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(3334,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)],[693]) ).

thf(529,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(2700,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(13,axiom,
    tc_Nat_Onat @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__mult) ).

thf(1038,plain,
    tc_Nat_Onat @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(744,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_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__mult__le) ).

thf(3554,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_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(657,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(3202,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)],[657]) ).

thf(236,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(1785,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)],[236]) ).

thf(730,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(3475,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)],[730]) ).

thf(75,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(1235,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)],[75]) ).

thf(102,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring) ).

thf(1317,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(30,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(1085,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)],[30]) ).

thf(902,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(4164,plain,
    tc_Nat_Onat @ class_Groups_Oordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(514,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(2656,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)],[514]) ).

thf(612,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(3019,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)],[612]) ).

thf(544,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(2747,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)],[544]) ).

thf(20,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(1053,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)],[20]) ).

thf(162,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(1499,plain,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero )
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(979,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oplus ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oplus) ).

thf(4409,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ocomm__monoid__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oplus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(987,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(4437,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)],[987]) ).

thf(307,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(1987,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)],[307]) ).

thf(218,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(1716,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)],[218]) ).

thf(897,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(4145,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)],[897]) ).

thf(414,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(2337,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)],[414]) ).

thf(813,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(3801,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)],[813]) ).

thf(380,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(2219,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)],[380]) ).

thf(305,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(1980,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)],[305]) ).

thf(279,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(1901,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)],[279]) ).

thf(743,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(3544,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)],[743]) ).

thf(459,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(2486,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)],[459]) ).

thf(925,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(4241,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)],[925]) ).

thf(478,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(2545,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)],[478]) ).

thf(761,axiom,
    tc_Nat_Onat @ class_Groups_Ozero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ozero) ).

thf(3623,plain,
    tc_Nat_Onat @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(746,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(3564,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)],[746]) ).

thf(650,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(3174,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)],[650]) ).

thf(323,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(2044,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)],[323]) ).

thf(224,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(1738,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)],[224]) ).

thf(615,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(3041,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)],[615]) ).

thf(438,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(2411,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)],[438]) ).

thf(176,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(1543,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)],[176]) ).

thf(768,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(3652,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)],[768]) ).

thf(485,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(2558,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)],[485]) ).

thf(700,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(3356,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)],[700]) ).

thf(1007,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(4488,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__1 )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Oone ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(493,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(2579,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)],[493]) ).

thf(227,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(1749,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)],[227]) ).

thf(185,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(1585,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)],[185]) ).

thf(918,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(4225,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)],[918]) ).

thf(593,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(2939,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)],[593]) ).

thf(859,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(3961,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)],[859]) ).

thf(121,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(1372,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)],[121]) ).

thf(703,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(3366,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)],[703]) ).

thf(495,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(2584,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)],[495]) ).

thf(981,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(4414,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)],[981]) ).

thf(975,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(4392,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)],[975]) ).

thf(77,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(1240,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)],[77]) ).

thf(326,axiom,
    tc_Int_Oint @ class_Rings_Oordered__ring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring) ).

thf(2056,plain,
    tc_Int_Oint @ class_Rings_Oordered__ring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(118,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(1358,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)],[118]) ).

thf(386,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(2247,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)],[386]) ).

thf(596,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(2948,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)],[596]) ).

thf(781,axiom,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__add) ).

thf(3697,plain,
    tc_Nat_Onat @ class_Groups_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(442,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(2426,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)],[442]) ).

thf(894,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(4133,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)],[894]) ).

thf(603,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(2992,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)],[603]) ).

thf(423,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(2357,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)],[423]) ).

thf(297,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(1955,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)],[297]) ).

thf(567,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(2843,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)],[567]) ).

thf(886,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(4106,plain,
    tc_Nat_Onat @ class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(164,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(1506,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)],[164]) ).

thf(881,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Groups_Ocomm__monoid__add) ).

thf(4093,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Groups_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(1,conjecture,
    ( ( v_p
      = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
    & ( v_a
      = ( t_a @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).

thf(2,negated_conjecture,
    ~ ( ( v_p
        = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      & ( v_a
        = ( t_a @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(1008,plain,
    ~ ( ( v_p
        = ( t_a @ tc_Polynomial_Opoly @ c_Groups_Ozero__class_Ozero ) )
      & ( v_a
        = ( t_a @ c_Groups_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(270,axiom,
    tc_Int_Oint @ class_Groups_Ozero,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ozero) ).

thf(1881,plain,
    tc_Int_Oint @ class_Groups_Ozero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(560,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(2807,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)],[560]) ).

thf(88,axiom,
    tc_Int_Oint @ class_Groups_Ouminus,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ouminus) ).

thf(1272,plain,
    tc_Int_Oint @ class_Groups_Ouminus,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(643,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(3156,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)],[643]) ).

thf(291,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(1940,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)],[291]) ).

thf(681,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(3284,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)],[681]) ).

thf(725,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(3452,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)],[725]) ).

thf(903,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(4165,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)],[903]) ).

thf(252,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(1817,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)],[252]) ).

thf(404,axiom,
    tc_Int_Oint @ class_Rings_Oidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oidom) ).

thf(2310,plain,
    tc_Int_Oint @ class_Rings_Oidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(378,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(2215,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)],[378]) ).

thf(996,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(4463,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)],[996]) ).

thf(144,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__imp__0__less) ).

thf(1453,plain,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(316,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(2013,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)],[316]) ).

thf(587,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(2924,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)],[587]) ).

thf(853,axiom,
    tc_Nat_Onat @ class_Power_Opower,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Power_Opower) ).

thf(3946,plain,
    tc_Nat_Onat @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(938,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(4278,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)],[938]) ).

thf(256,axiom,
    tc_Int_Oint @ class_Orderings_Oord,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Oord) ).

thf(1839,plain,
    tc_Int_Oint @ class_Orderings_Oord,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(760,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(3620,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)],[760]) ).

thf(285,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__eq__less__trans) ).

thf(1918,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(204,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(1649,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)],[204]) ).

thf(55,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(1157,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)],[55]) ).

thf(254,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(1834,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)],[254]) ).

thf(283,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(1913,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)],[283]) ).

thf(653,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(3188,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)],[653]) ).

thf(371,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(2194,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)],[371]) ).

thf(418,axiom,
    tc_Int_Oint @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ono__zero__divisors) ).

thf(2347,plain,
    tc_Int_Oint @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(135,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__0,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__0) ).

thf(1412,plain,
    tc_Int_Oint @ class_Rings_Ocomm__semiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(849,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(3928,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)],[849]) ).

thf(910,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(4185,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)],[910]) ).

thf(543,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(2743,plain,
    ! [A: $i] :
      ( ( A @ c_Nat_OSuc )
     != ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(461,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(2490,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)],[461]) ).

thf(370,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(2187,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)],[370]) ).

thf(63,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B != A )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__neq__implies__less) ).

thf(1189,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B != A )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(332,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__leI) ).

thf(2071,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
     => ( A @ ( B @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(515,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(2658,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)],[515]) ).

thf(549,axiom,
    tc_Nat_Onat @ class_Groups_Ocancel__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__semigroup__add) ).

thf(2775,plain,
    tc_Nat_Onat @ class_Groups_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(947,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(4310,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)],[947]) ).

thf(247,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_dvd__diffD1) ).

thf(1807,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) )
       => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Rings_Odvd__class_Odvd ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(582,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(2907,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)],[582]) ).

thf(891,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(4121,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)],[891]) ).

thf(110,axiom,
    tc_Int_Oint @ class_Groups_Oab__semigroup__add,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__add) ).

thf(1341,plain,
    tc_Int_Oint @ class_Groups_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(388,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( 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__eq ) ) )
       => ( ( C @ ( 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 @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_q__neg__lemma) ).

thf(2251,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( B @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( 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__eq ) ) )
       => ( ( C @ ( 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 @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(352,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(2135,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)],[352]) ).

thf(483,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(2554,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)],[483]) ).

thf(155,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
      <=> ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
          & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__fun__def) ).

thf(1481,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) )
         => ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless ) ) )
         => ( ~ ( B @ ( A @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) )
            & ( A @ ( B @ ( D @ ( C @ tc_fun ) @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(764,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(3630,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)],[764]) ).

thf(600,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(2971,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)],[600]) ).

thf(955,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(4338,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Ozero )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(666,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(3235,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)],[666]) ).

thf(704,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(3368,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)],[704]) ).

thf(180,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__antisym) ).

thf(1558,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(993,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1__strict ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict) ).

thf(4451,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1__strict ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(90,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(1276,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)],[90]) ).

thf(496,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(2591,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)],[496]) ).

thf(187,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(1590,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)],[187]) ).

thf(831,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(3864,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)],[831]) ).

thf(127,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
             => ( C = A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__eq__imp__eq__base) ).

thf(1385,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( ( B @ ( C @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) )
          = ( B @ ( A @ ( D @ c_Power_Opower__class_Opower @ hAPP ) @ hAPP ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
             => ( C = A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(497,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(2594,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)],[497]) ).

thf(431,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(2389,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)],[431]) ).

thf(153,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(1476,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)],[153]) ).

thf(809,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(3789,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)],[809]) ).

thf(571,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(2855,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)],[571]) ).

thf(784,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( A @ ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__monom__le) ).

thf(3709,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( A @ ( A @ ( B @ ( C @ c_Polynomial_Omonom ) ) @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(197,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__eq__le__trans) ).

thf(1629,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Orderings_Oord )
     => ( ( C = B )
       => ( ( A @ ( B @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( A @ ( C @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

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

thf(2004,plain,
    tc_Int_Oint @ class_Divides_Osemiring__div,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(473,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__mod__geq) ).

thf(2526,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( B @ ( A @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) )
        = ( B @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(807,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(3780,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)],[807]) ).

thf(449,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(2447,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)],[449]) ).

thf(691,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(3324,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)],[691]) ).

thf(941,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(4293,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)],[941]) ).

thf(835,axiom,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring) ).

thf(3879,plain,
    tc_Nat_Onat @ class_Rings_Ocomm__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(216,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(1706,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)],[216]) ).

thf(977,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1) ).

thf(4404,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(407,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(2315,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)],[407]) ).

thf(802,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(3761,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)],[802]) ).

thf(551,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(2779,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)],[551]) ).

thf(440,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(2421,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)],[440]) ).

thf(315,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(2009,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)],[315]) ).

thf(909,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(4182,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)],[909]) ).

thf(433,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__le__add2) ).

thf(2399,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(322,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(2041,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)],[322]) ).

thf(675,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(3267,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)],[675]) ).

thf(944,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ouminus ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ouminus) ).

thf(4299,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ouminus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(751,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(3598,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)],[751]) ).

thf(453,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(2462,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(27,axiom,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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_inverse__positive__iff__positive) ).

thf(1077,plain,
    ! [A: $i,B: $i] :
      ( ( B @ class_Fields_Olinordered__field__inverse__zero )
     => ( ( ( A @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( B @ c_Groups_Ozero__class_Ozero @ ( B @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( A @ ( B @ c_Rings_Oinverse__class_Oinverse ) @ ( 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)],[27]) ).

thf(361,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(2162,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)],[361]) ).

thf(810,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(3793,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)],[810]) ).

thf(368,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(2183,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)],[368]) ).

thf(580,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(2902,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)],[580]) ).

thf(347,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( C @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
           => ( D @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_unique__quotient__lemma__neg) ).

thf(2120,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( C @ ( E @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
           => ( D @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(383,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 ) ) )
    <=> ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__add1__eq__le) ).

thf(2230,plain,
    ! [A: $i,B: $i] :
      ( ( ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( 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 ) ) )
       => ( A @ ( B @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(320,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(2036,plain,
    tc_Int_Oint @ class_Groups_Oordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(99,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(1309,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)],[99]) ).

thf(276,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(1893,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)],[276]) ).

thf(398,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(2294,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)],[398]) ).

thf(541,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( B @ ( 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__left__le__one__le) ).

thf(2739,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Rings_Olinordered__idom )
     => ( ( B @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( A @ ( C @ c_Groups_Ozero__class_Ozero @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( C @ c_Groups_Oone__class_Oone @ ( A @ ( C @ c_Orderings_Oord__class_Oless__eq ) ) )
           => ( B @ ( 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)],[541]) ).

thf(883,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(4098,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)],[883]) ).

thf(201,axiom,
    tc_Int_Oint @ class_Rings_Ocomm__ring__1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring__1) ).

thf(1642,plain,
    tc_Int_Oint @ class_Rings_Ocomm__ring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(165,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mod__less__eq__dividend) ).

thf(1512,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Divides_Odiv__class_Omod ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(401,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(2302,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)],[401]) ).

thf(219,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(1725,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)],[219]) ).

thf(665,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(3232,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)],[665]) ).

thf(35,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(1097,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)],[35]) ).

thf(177,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__or__eq__imp__le) ).

thf(1546,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
        | ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
     => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(125,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( 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 ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__mono) ).

thf(1380,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Olinordered__semidom )
     => ( ( B @ ( C @ ( D @ c_Orderings_Oord__class_Oless ) ) )
       => ( ( C @ ( D @ c_Groups_Ozero__class_Ozero @ ( D @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( ( A @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( 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 ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(597,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(2951,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)],[597]) ).

thf(267,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(1876,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)],[267]) ).

thf(81,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(1248,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)],[81]) ).

thf(646,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(3164,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)],[646]) ).

thf(379,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(2218,plain,
    tc_Int_Oint @ class_Rings_Oring__1__no__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(893,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(4125,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)],[893]) ).

thf(48,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(1133,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)],[48]) ).

thf(211,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(1678,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)],[211]) ).

thf(737,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(3511,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)],[737]) ).

thf(961,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(4356,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)],[961]) ).

thf(862,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(3997,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)],[862]) ).

thf(73,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(1229,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)],[73]) ).

thf(752,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(3601,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)],[752]) ).

thf(694,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(3337,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)],[694]) ).

thf(209,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(1673,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)],[209]) ).

thf(509,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(2636,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)],[509]) ).

thf(857,axiom,
    tc_Nat_Onat @ class_Rings_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring) ).

thf(3956,plain,
    tc_Nat_Onat @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(867,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(4029,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)],[867]) ).

thf(251,axiom,
    tc_Int_Oint @ class_Rings_Osemiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring) ).

thf(1816,plain,
    tc_Int_Oint @ class_Rings_Osemiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(814,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(3804,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)],[814]) ).

thf(864,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(4007,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)],[864]) ).

thf(181,axiom,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__le__self) ).

thf(1561,plain,
    ! [A: $i,B: $i] : ( B @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(58,axiom,
    ! [A: $i] :
      ( A
     != ( A @ c_Nat_OSuc ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_n__not__Suc__n) ).

thf(1172,plain,
    ! [A: $i] :
      ( A
     != ( A @ c_Nat_OSuc ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(238,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(1789,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)],[238]) ).

thf(95,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(1288,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)],[95]) ).

thf(275,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(1891,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)],[275]) ).

thf(480,axiom,
    tc_Int_Oint @ class_Rings_Oring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring) ).

thf(2549,plain,
    tc_Int_Oint @ class_Rings_Oring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(141,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(1436,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)],[141]) ).

thf(262,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__diff__conv) ).

thf(1859,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(946,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(4303,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)],[946]) ).

thf(840,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(3897,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)],[840]) ).

thf(939,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(4281,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)],[939]) ).

thf(679,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(3277,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)],[679]) ).

thf(214,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(1693,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)],[214]) ).

thf(879,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(4083,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)],[879]) ).

thf(865,axiom,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Osemiring__0 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',clrel_Rings_Ocomm__semiring__0__Rings_Osemiring__0) ).

thf(4024,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Ocomm__semiring__0 )
     => ( A @ class_Rings_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(491,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(2572,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)],[491]) ).

thf(584,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(2913,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)],[584]) ).

thf(339,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(2091,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)],[339]) ).

thf(435,axiom,
    ! [A: $i] :
      ~ ( A @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__n__not__le__n) ).

thf(2404,plain,
    ! [A: $i] :
      ~ ( A @ ( A @ c_Nat_OSuc @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(989,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(4442,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)],[989]) ).

thf(869,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(4034,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)],[869]) ).

thf(804,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(3765,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)],[804]) ).

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

thf(2223,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( ( A = B )
         => ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ) )
        & ( ( B @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(556,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(2795,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)],[556]) ).

thf(152,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__assoc2) ).

thf(1473,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( A @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) )
        = ( A @ ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(503,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(2619,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)],[503]) ).

thf(24,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(1065,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)],[24]) ).

thf(415,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(2340,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)],[415]) ).

thf(56,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( 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 ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__one__le__iff__zero__less) ).

thf(1165,plain,
    ! [A: $i] :
      ( ( ( A @ ( 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__eq ) ) ) )
      & ( ( A @ ( tc_Int_Oint @ c_Groups_Oone__class_Oone @ ( 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 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(113,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(1346,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)],[113]) ).

thf(192,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(1610,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)],[192]) ).

thf(668,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(3244,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)],[668]) ).

thf(46,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(1129,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)],[46]) ).

thf(83,axiom,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__refl) ).

thf(1261,plain,
    ! [A: $i] : ( A @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(620,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(3057,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)],[620]) ).

thf(701,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(3360,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)],[701]) ).

thf(833,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(3869,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)],[833]) ).

thf(455,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(2467,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)],[455]) ).

thf(304,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(1976,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)],[304]) ).

thf(467,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__diff__assoc) ).

thf(2507,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) )
        = ( C @ ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(976,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(4395,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)],[976]) ).

thf(916,axiom,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ominus ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ominus) ).

thf(4221,plain,
    ! [A: $i] :
      ( ( A @ class_Groups_Oab__group__add )
     => ( A @ tc_Polynomial_Opoly @ class_Groups_Ominus ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(353,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(2138,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)],[353]) ).

thf(33,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(1093,plain,
    ! [A: $i] :
      ( ( A @ class_Rings_Olinordered__idom )
     => ( A @ tc_Polynomial_Opoly @ class_Rings_Olinordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(421,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(2353,plain,
    ! [A: $i] : ( A @ c_Nat_OSuc @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(561,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(2810,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)],[561]) ).

thf(540,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(2731,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)],[540]) ).

thf(599,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ! [D: $i] :
            ( ( D @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( ( D @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
       => ( B @ ( A @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_degree__le) ).

thf(2966,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C @ class_Groups_Ozero )
     => ( ! [D: $i] :
            ( ( D @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
           => ( ( D @ ( A @ ( C @ c_Polynomial_Ocoeff ) @ hAPP ) )
              = ( C @ c_Groups_Ozero__class_Ozero ) ) )
       => ( B @ ( A @ ( C @ c_Polynomial_Odegree ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(444,axiom,
    ! [A: $i] :
      ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( A
        = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__0__eq) ).

thf(2430,plain,
    ! [A: $i] :
      ( ( ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) )
       => ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( A
          = ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(86,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(1268,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)],[86]) ).

thf(499,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(2600,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)],[499]) ).

thf(970,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(4382,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)],[970]) ).

thf(753,axiom,
    tc_Nat_Onat @ class_Rings_Ono__zero__divisors,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ono__zero__divisors) ).

thf(3605,plain,
    tc_Nat_Onat @ class_Rings_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(715,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(3411,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)],[715]) ).

thf(174,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(1537,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)],[174]) ).

thf(1005,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(4484,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)],[1005]) ).

thf(732,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(3489,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)],[732]) ).

thf(793,axiom,
    tc_Nat_Onat @ class_Groups_Oone,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oone) ).

thf(3740,plain,
    tc_Nat_Onat @ class_Groups_Oone,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(269,axiom,
    tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).

thf(1880,plain,
    tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(876,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(4071,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)],[876]) ).

thf(711,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(3394,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)],[711]) ).

thf(131,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ? [C: $i] :
          ( A
          = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__iff__add) ).

thf(1399,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ? [C: $i] :
            ( A
            = ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(846,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(3918,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)],[846]) ).

thf(392,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
    <=> ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
       => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__cancel2) ).

thf(2270,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
       => ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
      & ( ( B @ ( A @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( C @ ( tc_Nat_Onat @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( B @ ( tc_Nat_Onat @ c_Groups_Ozero__class_Ozero @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless ) ) )
         => ( A @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(798,axiom,
    tc_Nat_Onat @ class_Rings_Oordered__semiring,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__semiring) ).

thf(3753,plain,
    tc_Nat_Onat @ class_Rings_Oordered__semiring,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(108,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(1331,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)],[108]) ).

thf(856,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(3950,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)],[856]) ).

thf(848,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(3924,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)],[848]) ).

thf(293,axiom,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__SucI) ).

thf(1945,plain,
    ! [A: $i,B: $i] :
      ( ( A @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( A @ c_Nat_OSuc @ ( B @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(573,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(2866,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)],[573]) ).

thf(328,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( E @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( E @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
           => ( B @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_unique__quotient__lemma) ).

thf(2059,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( A @ ( B @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( C @ ( D @ ( E @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( tc_Int_Oint @ c_Groups_Ozero__class_Ozero @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) )
       => ( ( E @ ( C @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
         => ( ( E @ ( A @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless ) ) )
           => ( B @ ( D @ ( tc_Int_Oint @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(984,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(4423,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)],[984]) ).

thf(874,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(4056,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)],[874]) ).

thf(906,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(4174,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)],[906]) ).

thf(748,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(3575,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)],[748]) ).

thf(265,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(1870,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)],[265]) ).

thf(728,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(3461,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)],[728]) ).

thf(104,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(1320,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)],[104]) ).

thf(525,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ 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_Rings_Odvd__class_Odvd ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__imp__power__dvd) ).

thf(2690,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D @ class_Rings_Ocomm__semiring__1 )
     => ( ( B @ ( C @ ( tc_Nat_Onat @ 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_Rings_Odvd__class_Odvd ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(605,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(2998,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)],[605]) ).

thf(445,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(2438,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)],[445]) ).

thf(158,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(1490,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)],[158]) ).

thf(12,axiom,
    tc_Int_Oint @ class_Groups_Omonoid__mult,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__mult) ).

thf(1037,plain,
    tc_Int_Oint @ class_Groups_Omonoid__mult,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(399,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 @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( ( C @ ( B @ ( 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__eq ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_self__quotient__aux1) ).

thf(2297,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 @ ( C @ ( tc_Int_Oint @ c_Groups_Otimes__class_Otimes @ hAPP ) @ hAPP ) @ ( B @ ( tc_Int_Oint @ c_Groups_Oplus__class_Oplus ) ) ) )
       => ( ( C @ ( B @ ( 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__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(948,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(4313,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)],[948]) ).

thf(687,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(3316,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)],[687]) ).

thf(420,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(2351,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)],[420]) ).

thf(31,axiom,
    tc_Nat_Onat @ class_Rings_Olinordered__semidom,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semidom) ).

thf(1088,plain,
    tc_Nat_Onat @ class_Rings_Olinordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(67,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(1204,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)],[67]) ).

thf(250,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(1812,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)],[250]) ).

thf(342,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(2106,plain,
    tc_Int_Oint @ class_Rings_Olinordered__comm__semiring__strict,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(338,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
      <=> ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__diff__conv2) ).

thf(2087,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( B @ ( C @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
     => ( ( ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) )
        & ( ( C @ ( B @ ( tc_Nat_Onat @ c_Groups_Ominus__class_Ominus ) ) @ ( A @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) )
         => ( B @ ( C @ ( A @ ( tc_Nat_Onat @ c_Groups_Oplus__class_Oplus ) ) @ ( tc_Nat_Onat @ c_Orderings_Oord__class_Oless__eq ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(851,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(3933,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)],[851]) ).

thf(617,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(3047,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)],[617]) ).

thf(456,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(2470,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)],[456]) ).

thf(365,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(2177,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)],[365]) ).

thf(513,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(2655,plain,
    tc_Int_Oint @ class_Groups_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(4543,plain,
    $false,
    inference(e,[status(thm)],[2199,3883,1633,3798,3666,1237,1899,1840,1083,2331,1950,2250,2822,4188,4201,2452,4447,1879,2094,1411,2612,1469,3053,4101,4262,1767,2263,3930,2921,3698,1315,3830,2787,2280,1110,3172,4037,1924,1882,2776,3153,3912,2671,1728,4479,1486,3680,4043,3534,3681,3990,1596,2104,4326,3498,1971,1793,1533,4234,3429,2686,4110,1999,3616,1342,3313,1063,1354,1458,1788,1426,4090,1655,4026,3044,2627,2168,1222,3717,2360,1095,2046,3502,1391,3777,3281,3140,2825,1623,1887,4386,3836,4343,2348,2723,1243,1127,2763,3915,4079,4266,1935,3947,4156,1803,3880,1904,1518,1265,1703,2531,4467,1709,3689,3342,3495,2501,4334,3125,3744,2126,3855,4238,1055,1409,3263,4439,2603,2916,3425,1792,1430,2698,1298,2708,3821,1323,3612,4275,4349,2185,1577,1455,2562,2488,4243,2703,1922,3099,1514,2049,1479,2373,2929,3657,1773,4390,2852,3866,1847,2713,4119,1379,3385,1013,4270,1741,3084,3734,4368,3014,1232,4253,3258,3322,2794,2996,1136,1828,1151,2291,2222,4411,4329,2324,1811,1450,2893,4297,2451,2236,3438,3525,2122,3290,3093,1283,2073,1051,4162,3537,4230,1215,1896,2483,3505,1119,2772,1351,2286,3327,2750,1242,2718,4475,4082,3378,3808,1796,1072,4040,3756,3226,1943,1201,3606,3211,2392,3435,2954,3229,4199,2910,2204,2473,1691,3720,3169,2364,1153,1470,2641,1074,1604,3852,1306,1783,4370,3767,2307,2083,4031,1736,2496,2790,2905,1868,2061,1636,1047,3679,1042,4434,2147,3022,1487,3371,3814,3471,2441,2673,3827,2428,3711,2560,2805,1676,2133,4449,3134,4284,1619,2150,2414,3250,3339,2015,2628,4053,3090,1555,2872,2068,3144,2181,1027,1392,1106,3948,3307,3271,3166,2445,4295,3191,2464,1938,1032,1837,3269,3137,3750,3349,3556,3578,1178,4417,3050,2211,1407,2547,2313,4104,1281,2741,3033,3408,1910,2769,4316,3731,4123,2840,1762,4248,4486,1970,3102,2631,3078,3650,2638,3699,2300,3909,3782,1089,2498,1270,3122,1017,1953,2692,1498,1540,3518,4095,3119,3205,4219,3176,3087,1567,1329,2877,1361,3179,3846,3407,1182,1461,3714,2080,1842,4406,1791,3194,2108,1889,1155,2460,1493,2798,1028,4301,2023,1964,1659,3922,3441,3010,2551,1207,3723,3643,4444,2666,1612,4180,1039,2504,1263,4384,2681,2304,1131,1552,3819,1799,1947,1984,1599,3064,3005,2140,2370,2932,3486,3654,1588,1099,2836,1382,3354,2153,2900,2355,1192,3222,2006,3247,2862,3201,4223,1759,3147,1333,1318,2926,4352,3624,2883,1563,2057,2076,1466,2408,1515,3422,1996,3807,3824,3903,1857,1286,3321,2368,2662,1185,2423,2529,2253,3890,1273,2570,1927,3686,1374,1278,1906,2936,2729,3660,2001,4470,1754,4380,2556,4354,3257,1219,3607,4340,3375,3839,4359,3274,2524,4453,3372,4000,3741,3319,1916,1884,1125,1752,2016,1388,1535,3458,2716,3404,4268,3070,3811,2791,2525,3609,4159,2849,1652,3081,3758,4465,3900,2695,1226,2326,1312,1058,3594,2442,4060,3304,1464,1356,3532,1428,4251,1139,3796,1503,4171,3208,4236,3663,1865,2935,3002,1733,1765,3619,4273,1258,1638,4245,2510,2311,2521,2679,4228,2151,2084,1117,3419,1349,1603,2116,2945,1122,4388,1194,2346,2119,3352,3764,2493,1806,1011,3957,3683,1524,2669,4139,1144,1801,4154,2378,4374,3066,1376,1439,4107,1307,2197,4477,2180,3219,3747,2267,4332,1325,3451,1344,1022,2278,2974,4420,2605,2343,3668,2625,1731,4264,3298,2874,2542,2647,3150,2329,3893,2710,1112,2578,1688,1171,4232,2401,2835,1528,1176,2288,3142,2535,1364,1496,1102,2156,2207,4473,3627,2171,1044,2550,1245,4373,1643,3159,2064,1015,3241,2797,3016,2633,3363,2984,1973,4481,4426,1809,1873,2037,2124,3455,4336,2005,1413,1626,3009,1034,2239,1662,4136,3936,2107,2318,4456,3161,1420,1081,4177,1990,2684,2202,2333,2846,4113,2564,2597,1958,4168,2582,3571,3849,1049,2942,3391,3737,2715,3640,1961,2143,1647,3833,2175,1531,3988,1845,2513,2567,1743,4366,3260,3754,2407,2386,2720,3872,4377,3448,4190,3334,2700,1038,3554,3202,1785,3475,1235,1317,1085,4164,2656,3019,2747,1053,1499,4409,4437,1987,1716,4145,2337,3801,2219,1980,1901,3544,2486,4241,2545,3623,3564,3174,2044,1738,3041,2411,1543,3652,2558,3356,4488,2579,1749,1585,4225,2939,3961,1372,3366,2584,4414,4392,1240,2056,1358,2247,2948,3697,2426,4133,2992,2357,1955,2843,4106,1506,4093,1008,1881,2807,1272,3156,1940,3284,3452,4165,1817,2310,2215,4463,1453,2013,2924,3946,4278,1839,3620,1918,1649,1157,1834,1913,3188,2194,2347,1412,3928,4185,2743,2490,2187,1189,2071,2658,2775,4310,1807,2907,4121,1341,2251,2135,2554,1481,3630,2971,4338,3235,3368,1558,4451,1276,2591,1590,3864,1385,2594,2389,1476,3789,2855,3709,1629,2004,2526,3780,2447,3324,4293,3879,1706,4404,2315,3761,2779,2421,2009,4182,2399,2041,3267,4299,3598,2462,1077,2162,3793,2183,2902,2120,2230,2036,1309,1893,2294,2739,4098,1642,1512,2302,1725,3232,1097,1546,1380,2951,1876,1248,3164,2218,4125,1133,1678,3511,4356,3997,1229,3601,3337,1673,2636,3956,4029,1816,3804,4007,1561,1172,1789,1288,1891,2549,1436,1859,4303,3897,4281,3277,1693,4083,4024,2572,2913,2091,2404,4442,4034,3765,2223,2795,1473,2619,1065,2340,1165,1346,1610,3244,1129,1261,3057,3360,3869,2467,1976,2507,4395,4221,2138,1093,2353,2810,2731,2966,2430,1268,2600,4382,3605,3411,1537,4484,3489,3740,1880,4071,3394,1399,3918,2270,3753,1331,3950,3924,1945,2866,2059,4423,4056,4174,3575,1870,3461,1320,2690,2998,2438,1490,1037,2297,4313,3316,2351,1088,1204,1812,2106,2087,3933,3047,2470,2177,2655]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : SWW186+1 : TPTP v9.3.1. Released v5.2.0.
% 0.00/0.09  % 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
% 0.14/0.40  % Computer : n009.cluster.edu
% 0.14/0.40  % Model    : x86_64 x86_64
% 0.14/0.40  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.40  % Memory   : 8046.5625MB
% 0.14/0.40  % OS       : Linux 6.8.0-71-generic
% 0.14/0.40  % CPULimit : 300
% 0.14/0.40  % WCLimit  : 300
% 0.14/0.40  % DateTime : Sat Sep 26 15:26:44 UTC 2026
% 0.14/0.40  % CPUTime  : 
% 0.14/0.40  Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.79/1.02  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 2.47/1.75  % [INFO] 	 Parsing done (727ms). 
% 2.47/1.78  % [INFO] 	 Running in sequential loop mode. 
% 3.42/2.24  % [INFO] 	 eprover registered as external prover. 
% 3.42/2.26  % [INFO] 	 Scanning for conjecture ... 
% 4.04/2.54  % [INFO] 	 Found a conjecture (or negated_conjecture) and 1181 axioms. Running axiom selection ... 
% 5.85/3.28  % [INFO] 	 Axiom selection finished. Selected 1005 axioms (removed 176 axioms). 
% 7.75/4.09  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 8.06/4.19  % [INFO] 	 Type checking passed. 
% 8.06/4.19  % [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 ... 
% 32.50/14.38  % External prover 'e' found a proof!
% 32.50/14.38  % [INFO] 	 Killing All external provers ... 
% 32.50/14.38  % Time passed: 13866ms (effective reasoning time: 12589ms)
% 32.50/14.38  % 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)>
% 32.50/14.39  % Axioms used in derivation (1005): fact_add__le__less__mono, fact_dvd_Oord__less__eq__trans, fact_minus__monom, fact_le__SucI, fact_smult__minus__left, arity_Polynomial__Opoly__Rings_Osemiring__0, fact_dvd__imp__degree__le, fact_nat_Osimps_I3_J, fact_le__imp__diff__is__add, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J, fact_dvd_Oless__trans, fact_zdvd__zmod, fact_incr__mult__lemma, fact_zle__refl, fact_minus__mult__left, fact_Nat_Oadd__0__right, fact_degree__pcompose__le, fact_one__dvd, fact_real__squared__diff__one__factored, 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_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_mult__1__left, fact_diff__0__eq__0, fact_dvd__mult__cancel__left, fact_zdvd__not__zless, fact_neg__equal__0__iff__equal, fact_diff__add__inverse, fact_dvdI, fact_le__eq__less__or__eq, fact_minus__mult__right, fact_degree__smult__le, fact_odd__nonzero, fact_trans__less__add1, fact_q__pos__lemma, fact_add__poly__code_I2_J, fact_inverse__1, fact_pdivmod__rel__unique__div, fact_le__degree, fact_order, fact_diff__0__right, fact_nat__add__left__commute, fact_mult__nonpos__nonpos, fact_dvd_Oeq__iff, fact_power__power__power, fact_degree__add__eq__right, tfree_0, fact_inf__period_I4_J, fact_zdvd__antisym__nonneg, fact_power__inject__base, arity_Nat__Onat__Rings_Ono__zero__divisors, fact_nat__less__cases, fact_one__is__add, fact_diff__le__mono2, arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add, fact_termination__basic__simps_I5_J, arity_Polynomial__Opoly__Rings_Ocomm__semiring__0, fact_dvd_Oorder__trans, fact_zminus__zminus, fact_mult__nonneg__nonpos, fact_poly__eq__0__iff__dvd, fact_less__imp__le__nat, fact_mult__right__less__imp__less, fact_combine__common__factor, fact_poly__gcd__greatest, fact_n__less__m__mult__n, fact_inverse__inverse__eq, fact_unity__coeff__ex, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J, fact_Suc__mono, fact_less__nat__zero__code, fact_one__less__inverse__iff, arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le, 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_mod__mult__cong, fact_le__imp__0__less, fact_dvd__triv__right, fact_add__leD1, fact_split__mult__neg__le, fact_zpower__zmod, fact_dvd_Oless__asym_H, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, fact_inverse__add, fact_zero__less__power, arity_Nat__Onat__Rings_Ocomm__semiring__0, fact_inverse__negative__iff__negative, arity_Nat__Onat__Groups_Omonoid__mult, fact_power__increasing, clrel_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__add, fact_mult__0, fact_inverse__zero__imp__zero, fact_mod__mult__eq, fact_dvd_Oless__imp__not__less, fact_add__right__imp__eq, arity_Polynomial__Opoly__Rings_Oidom, fact_zero__less__diff, fact_nonzero__inverse__eq__imp__eq, fact_diff__diff__cancel, fact_mult__mono, fact_nat__add__right__cancel, fact_field__power__not__zero, fact_Suc__le__mono, 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_dvd__mod, fact_gr0__conv__Suc, arity_Polynomial__Opoly__Groups_Ominus, fact_coeff__eq__0, fact_poly__dvd__antisym, fact_zdvd__mult__cancel, fact_inverse__zero, fact_dvd_Oord__eq__less__trans, fact_mult__le__mono2, fact_Suc__diff__diff, fact_mod__diff__cong, fact_minus__add__cancel, fact_smult__smult, fact_zle__linear, fact_zero__le__one, arity_Int__Oint__Orderings_Oord, fact_nat__add__left__cancel__less, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J, fact_mod__mult__distrib2, fact_mod__less, fact_coeff__add, fact_le__add__iff2, help_c__fequal__2, fact_mult__eq__self__implies__10, fact_minus__pCons, fact_poly__mod__minus__left, fact_le__neq__implies__less, fact_dvd__diffD, arity_Polynomial__Opoly__Rings_Ocomm__ring, fact_degree__linear__power, fact_pow__divides__pow__nat, fact_poly__add, fact_degree__pCons__0, fact_order__1, fact_int__0__less__1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J, fact_linorder__neqE__linordered__idom, fact_mod__if, fact_synthetic__div__correct_H, fact_order__degree, fact_smult__1__left, fact_zadd__left__mono, fact_mod__mult__self4, arity_Int__Oint__Rings_Olinordered__semiring__1__strict, fact_add__le__imp__le__left, fact_mod__mult__self1, fact_sum__squares__ge__zero, fact_pow__divides__eq__nat, fact_dvd__reduce, fact_pdivmod__rel__by__0, fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J, fact_dvd__power__le, fact_mult__smult__right, fact_smult__pCons, fact_mod__by__1, arity_Polynomial__Opoly__Rings_Olinordered__semidom, fact_synthetic__div__0, arity_Int__Oint__Groups_Ozero, fact_zdvd__zmod__imp__zdvd, fact_power__commutes, fact_poly__0, fact_add__mono, fact_gr0I, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J, fact_add__0__iff, arity_Polynomial__Opoly__Orderings_Oord, fact_degree__1, fact_power__strict__increasing, fact_one__le__power, fact_even__less__0__iff, fact_self__quotient__aux1, 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_coeff__1, arity_Int__Oint__Rings_Olinordered__ring, fact_poly__smult, fact_le__funD, fact_add__right__mono, fact_Divides_Otransfer__nat__int__function__closures_I2_J, fact_le__fun__def, fact_minus__add, arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add, fact_power__minus, fact_neg__mod__sign, arity_Int__Oint__Groups_Ocancel__comm__monoid__add, fact_one__less__power, fact_mod__mult__left__eq, fact_nat__dvd__not__less, fact_leading__coeff__neq__0, arity_Nat__Onat__Rings_Osemiring__0, fact_diff__le__self, arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict, fact_mult__eq__1__iff, fact_smult__add__right, arity_fun__Groups_Ouminus, fact_degree__pCons__eq__if, arity_Nat__Onat__Rings_Oordered__comm__semiring, fact_power__eq__0__iff, fact_diff__Suc__eq__diff__pred, fact_less__add__eq__less, fact_nat__mult__1__right, fact_mod__1, fact_eq__iff__diff__eq__0, arity_Int__Oint__Rings_Oidom, fact_coeff__linear__power, fact_ord__less__eq__trans, fact_poly__gcd_Oassoc, fact_termination__basic__simps_I4_J, fact_le__iff__add, fact_one__reorient, fact_diff__commute, fact_minus__mult__commute, arity_Polynomial__Opoly__Groups_Omonoid__add, fact_division__ring__inverse__add, fact_add__lessD1, fact_add__le__mono1, fact_poly__gcd__monic, fact_less__eq__nat_Osimps_I1_J, fact_less__degree__imp, fact_dvd__eq__mod__eq__0, 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_synthetic__div__pCons, 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_Suc__leI, fact_poly__mult, fact_split__mult__pos__le, fact_offset__poly__pCons, fact_nat_Oinject, fact_neg__mod__bound, fact_mult__left__le__one__le, arity_Polynomial__Opoly__Power_Opower, arity_Int__Oint__Rings_Oring__no__zero__divisors, fact_not__add__less2, fact_dvd__trans, fact_less__or__eq__imp__le, fact_mod__mod__trivial, fact_Suc__less__SucD, fact_dvd_Oeq__refl, fact_power__strict__mono, fact_zadd__strict__right__mono, fact_q__neg__lemma, fact_minus__apply, fact_zdvd__reduce, fact_inverse__le__1__iff, fact_add__0__right, fact_unique__quotient__lemma__neg, fact_diff__less, fact_power__Suc2, fact_convex__bound__lt, fact_zpower__zadd__distrib, fact_le__add2, 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_mod__diff__right__eq, fact_Suc__lessD, fact_Suc__n__not__le__n, fact_pdivmod__rel__0__iff, fact_not__less__less__Suc__eq, fact_diff__def, fact_dvd__mult2, fact_nat__mult__1, fact_diff__is__0__eq_H, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J, fact_mult__right__mono__neg, fact_diff__pCons, 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_poly__gcd__dvd2, arity_Int__Oint__Power_Opower, fact_diff__0, fact_add__left__imp__eq, fact_poly__gcd__minus__right, fact_zadd__0, fact_zpower__zpower, fact_add__leD2, fact_mult__mono_H, arity_Nat__Onat__Rings_Ocomm__semiring__1, fact_realpow__minus__mult, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, fact_dvd__pos__nat, fact_add__diff__assoc, arity_Nat__Onat__Power_Opower, fact_inverse__less__1__iff, fact_offset__poly__single, fact_ab__semigroup__mult__class_Omult__ac_I1_J, fact_not__less0, fact_mod__mult__self1__is__0, arity_Int__Oint__Rings_Osemiring__0, clrel_Rings_Ocomm__semiring__0__Groups_Ozero, fact_monom__Suc, fact_trans__less__add2, fact_poly__offset__poly, arity_Int__Oint__Groups_Ouminus, fact_less__Suc__eq__le, arity_Nat__Onat__Groups_Ocancel__comm__monoid__add, fact_mod__less__divisor, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J, fact_pcompose__0, fact_degree__pCons__le, fact_dvd_Oless__asym, fact_offset__poly__0, fact_zmod__simps_I3_J, fact_coeff__minus, fact_zmod__zero, fact_synthetic__div__correct, fact_not__one__less__zero, fact_degree__smult__eq, fact_one__le__mult__iff, fact_not__less__eq, fact_nat_Osimps_I2_J, 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_mult__left__less__imp__less, fact_smult__eq__0__iff, fact_pdivmod__rel__smult__right, arity_Int__Oint__Divides_Osemiring__div, fact_less__zeroE, fact_termination__basic__simps_I2_J, fact_minus__zero, fact_mult__le__cancel1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J, fact_nat__neq__iff, fact_nat__1__eq__mult__iff, fact_diff__mult__distrib2, fact_dvd_Ole__less, arity_Nat__Onat__Groups_Ominus, fact_degree__pCons__eq, fact_pCons__0__0, arity_Nat__Onat__Groups_Oplus, arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict, fact_power__less__imp__less__base, fact_add__mult__distrib2, fact_eq__add__iff1, arity_Int__Oint__Groups_Ocancel__ab__semigroup__add, fact_less__not__refl3, fact_mod__Suc, fact_add__right__cancel, fact_poly__pCons, 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, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J, fact_nat__add__left__cancel__le, arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult, arity_Int__Oint__Groups_Oplus, fact_nat__one__le__power, fact_one__less__mult, 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_zmult__assoc, fact_degree__le, arity_Int__Oint__Rings_Oordered__semiring, fact_dvd_Oord__le__eq__trans, clrel_Rings_Ocomm__semiring__0__Rings_Ocomm__semiring, fact_neg__0__equal__iff__equal, fact_zminus__zadd__distrib, fact_le__diff__conv, fact_one__neq__zero, arity_Polynomial__Opoly__Rings_Oordered__comm__semiring, fact_right__minus, fact_mult__left__mono, fact_le__Suc__eq, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J, fact_double__zero__sym, fact_mod__mult__mult1, fact_pdivmod__rel__smult__left, fact_zmult__1, fact_left__minus, fact_diff__cancel, fact_power__decreasing, fact_le__add__iff1, fact_monom__eq__0__iff, 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_offset__poly__eq__0__lemma, arity_Polynomial__Opoly__Groups_Oab__group__add, fact_poly__pcompose, arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add, fact_le__imp__power__dvd, arity_Nat__Onat__Groups_Ocomm__monoid__add, fact_nat__mult__le__cancel1, fact_le__add__diff__inverse2, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J, arity_Int__Oint__Rings_Ocomm__ring, fact_nat__mult__assoc, fact_equation__minus__iff, fact_le__funE, fact_poly__gcd__dvd1, 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, fact_Suc__le__eq, arity_Int__Oint__Groups_Ogroup__add, fact_add__diff__cancel, fact_mult__cancel2, arity_Polynomial__Opoly__Groups_Ouminus, 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_mod__neg__neg__trivial, fact_smult__add__left, fact_less__eq__Suc__le, fact_mod__le__divisor, fact_uminus__apply, help_c__fequal__1, arity_Polynomial__Opoly__Divides_Osemiring__div, arity_Int__Oint__Rings_Oring__1__no__zero__divisors, fact_zmult__commute, fact_comm__mult__strict__left__mono, fact_dvd__diffD1, arity_Nat__Onat__Orderings_Oord, fact_equal__neg__zero, 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_inverse__nonnegative__iff__nonnegative, fact_mult__nonneg__nonpos2, fact_less__add__Suc2, arity_Polynomial__Opoly__Groups_Omonoid__mult, fact_dvd__power__same, fact_less__imp__diff__less, fact_power__add, fact_inverse__eq__imp__eq, fact_pdivmod__rel__unique, fact_nat__mult__eq__cancel1, fact_pCons__eq__iff, fact_dvd_Oless__not__sym, fact_degree__add__less, fact_mod__by__0, fact_diff__add__cancel, arity_Int__Oint__Groups_Oab__group__add, fact_ord__le__eq__trans, fact_pdivmod__rel__by__0__iff, fact_minus__minus, fact_dvd__mult__cancel1, fact_mod__diff__eq, fact_mod__mult__right__eq, fact_smult__0__left, fact_minus__add__distrib, fact_diffs0__imp__equal, fact_dvd__diff, arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add, arity_Nat__Onat__Groups_Ozero, fact_pCons__eq__0__iff, 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_linorder__neqE__nat, fact_mult__less__mono2, fact_smult__monom, fact_zmod__zminus__zminus, fact_diff__diff__left, fact_power__inject__exp, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J, fact_nat__mult__less__cancel1, fact_zadd__left__commute, fact_diff__diff__right, fact_add__less__cancel__right, fact_Suc__mult__cancel1, arity_Nat__Onat__Groups_Ocomm__monoid__mult, arity_Int__Oint__Groups_Omonoid__mult, fact_diff__add__assoc2, fact_not__add__less1, fact_neg__mod__conj, clrel_Rings_Ocomm__semiring__0__Groups_Omonoid__add, arity_Int__Oint__Int_Oring__char__0, fact_expand__poly__eq, fact_unique__quotient__lemma, fact_dvd__add, 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_mult__poly__add__left, fact_Suc__le__lessD, fact_inverse__nonzero__iff__nonzero, fact_mod__add__self2, fact_mod__geq, fact_power__increasing__iff, clrel_Rings_Ocomm__semiring__0__Groups_Ocomm__monoid__add, fact_zless__le, arity_fun__Orderings_Oord, fact_ext, fact_mod__self, fact_poly__gcd_Oleft__commute, arity_Polynomial__Opoly__Rings_Ocomm__semiring__1, fact_zless__imp__add1__zle, fact_poly__gcd_Ocommute, fact_eq__diff__iff, fact_degree__add__le, fact_zero__le__power, fact_Suc__neq__Zero, fact_power__strict__increasing__iff, arity_Polynomial__Opoly__Rings_Omult__zero, fact_inverse__eq__1__iff, fact_poly__gcd__code, fact_Suc__inject, fact_dvd__poly__gcd__iff, fact_minus__dvd__iff, fact_less__diff__conv, 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_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_add__eq__self__zero, fact_diff__less__Suc, fact_mod__poly__less, fact_zmod__eq__0__iff, fact_uminus__dvd__conv_I2_J, fact_mult__le__cancel2, arity_Int__Oint__Rings_Omult__zero, fact_zmod__le__nonneg__dividend, fact_dvd_Oless__imp__le, fact_mult__le__mono, fact_dvd__imp__le, fact_nat__mult__dvd__cancel1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J, arity_Int__Oint__Rings_Oring, fact_zminus__0, fact_power__less__imp__less__exp, fact_mult__is__0, fact_nat__add__left__cancel, arity_Polynomial__Opoly__Groups_Oplus, fact_n__not__Suc__n, fact_zminus__zmod, fact_nat__le__linear, fact_mult__zero__right, fact_mod__add__left__eq, fact_poly__rec__0, fact_mod__diff__left__eq, arity_Nat__Onat__Rings_Omult__zero, fact_inverse__positive__iff__positive, fact_inverse__mult__distrib, 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_coeff__mult__degree__sum, fact_nat__0__less__mult__iff, fact_add__gr__0, arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add, fact_add__diff__assoc2, fact_poly__gcd__1__right, clrel_Rings_Ocomm__semiring__0__Rings_Osemiring__0, fact_dvd__1__left, arity_Polynomial__Opoly__Rings_Ozero__neq__one, fact_add__mult__distrib, fact_less__minus__self__iff, fact_power__dvd__imp__le, fact_le__add__diff, fact_pos__poly__total, arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict, fact_inverse__nonpositive__iff__nonpositive, fact_add__Suc, fact_diff__add__assoc, fact_dvd__triv__left, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J, clrel_Rings_Ocomm__semiring__0__Rings_Osemiring, fact_inverse__eq__iff__eq, fact_minus__unique, fact_poly__power, fact_diff__less__mono2, fact_dvd_Oless__imp__not__eq, arity_Nat__Onat__Rings_Oordered__cancel__semiring, fact_diff__mult__distrib, fact_int__0__neq__1, fact_dvd__diff__nat, fact_power__Suc, fact_zmod__zminus2__not__zero, fact_Suc__eq__plus1__left, fact_field__le__mult__one__interval, arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add, fact_left__inverse, fact_power__0, fact_power__0__left, fact_add_Ocomm__neutral, fact_add__less__imp__less__right, fact_le__Suc__ex__iff, arity_Nat__Onat__Rings_Odvd, arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le, fact_Zero__not__Suc, fact_dvd__mult__right, fact_gcd__lcm__complete__lattice__nat_Otop__greatest, fact_diff__less__mono, fact_Suc__leD, fact_zero__less__one, fact_power__le__dvd, 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, clrel_Rings_Ocomm__semiring__0__Groups_Oplus, fact_zmod__self, fact_nonzero__imp__inverse__nonzero, arity_Nat__Onat__Rings_Oordered__semiring, fact_mult__Suc, fact_less__trans__Suc, fact_le__minus__self__iff, fact_mod__mult__self3, fact_less__not__refl2, fact_poly__rec_Osimps, fact_poly__gcd__zero__iff, arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors, fact_poly__monom, fact_zmod__zmult1__eq, fact_mult__right__le__one__le, 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_Nat_Odiff__diff__eq, fact_le__refl, fact_nat__add__assoc, fact_eq__add__iff2, fact_zadd__zmult__distrib2, fact_le__add1, fact_mult__cancel1, fact_trans__le__add2, fact_zdvd__mono, fact_monom__eq__0, fact_less__add__Suc1, fact_add__is__0, fact_termination__basic__simps_I3_J, clrel_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__mult, fact_mult__diff__mult, fact_add__less__imp__less__left, fact_ab__diff__minus, fact_le__imp__less__Suc, fact_mod__less__eq__dividend, fact_mult__right__mono, arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add, conj_1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J, arity_Int__Oint__Rings_Olinordered__semiring, fact_coeff__smult, fact_smult__0__right, fact_leading__coeff__0__iff, fact_Suc__mult__le__cancel1, fact_ab__left__minus, fact_dvd__mod__iff, fact_less__fun__def, fact_One__nat__def, fact_minus__diff__eq, fact_comm__semiring__class_Odistrib, fact_square__eq__iff, fact_smult__dvd__iff, fact_mod__eq__0__iff, arity_Int__Oint__Rings_Oordered__ring, fact_mod__mult__mult2, fact_power_Opower_Opower__Suc, arity_Polynomial__Opoly__Rings_Oordered__ring, fact_diff__minus__eq__add, fact_power__mono, fact_Suc__eq__plus1, fact_mult__le__mono1, fact_diff__self__eq__0, fact_add__eq__0__iff, fact_minus__poly__code_I2_J, fact_degree__minus, fact_diff__add__0, fact_uminus__dvd__conv_I1_J, fact_dvd__minus__iff, 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_self__quotient__aux2, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J, fact_neg__equal__iff__equal, fact_degree__power__le, fact_realpow__two__disj, fact_inverse__minus__eq, arity_Polynomial__Opoly__Rings_Olinordered__semiring__1, fact_pos__poly__add, fact_coeff__pCons__Suc, fact_smult__dvd, fact_mult__nonpos__nonneg, fact_zmod__simps_I2_J, fact_Suc__diff__le, 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, arity_Int__Oint__Groups_Ocomm__monoid__mult, fact_add__left__mono, fact_ord__eq__less__trans, arity_Int__Oint__Rings_Oordered__comm__semiring, fact_zmult__zless__mono2, arity_Int__Oint__Groups_Oab__semigroup__add, fact_nat__dvd__1__iff__1, fact_zmod__simps_I1_J, arity_Int__Oint__Groups_Oone, fact_le__mod__geq, arity_Nat__Onat__Rings_Olinordered__semidom, fact_add__le__cancel__right, fact_power__Suc__0, fact_nat__mult__eq__1__iff, fact_not__less__eq__eq, fact_n__less__n__mult__m, 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_le0, fact_coeff__monom, fact_power__one, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J, fact_poly__diff, fact_mod__smult__left, fact_le__diff__conv2, fact_minus__mult__minus, fact_ex__least__nat__less, fact_crossproduct__eq, fact_monom__eq__iff, fact_square__eq__1__iff, fact_dvd_Oantisym, fact_le__trans, fact_mod__mod__cancel, fact_comm__mult__left__mono, fact_degree__mod__less, fact_inf__period_I3_J, arity_fun__Groups_Ominus, fact_gr__implies__not0, fact_dvd_Ole__neq__trans, fact_dvd__antisym, fact_power__less__power__Suc, arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors, fact_less__add__one, fact_nonzero__power__inverse, fact_Suc__less__eq, fact_add__leE, fact_zmod__zminus2, fact_dvd__smult__iff, arity_Int__Oint__Rings_Odvd, fact_poly__gcd__1__left, arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add, fact_diff__Suc__1, fact_add__less__le__mono, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J, fact_int__one__le__iff__zero__less, fact_degree__monom__eq, fact_poly__gcd_Osimps_I1_J, fact_zero__neq__one, fact_power__eq__imp__eq__base, fact_less__Suc__eq, 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, arity_Int__Oint__Rings_Oring__1, fact_right__minus__eq, fact_coeff__0, fact_zmult__zminus, fact_zdiv__mono2__lemma, arity_Polynomial__Opoly__Groups_Ogroup__add, fact_degree__mult__le, fact_zero__less__Suc, fact_eq__zero__or__degree__less, fact_plus__nat_Oadd__0, fact_mult_Ocomm__neutral, fact_dvd_Oless__le, fact_pow__divides__eq__int, fact_le__SucE, fact_dvd_Oless__le__trans, fact_nat__power__eq__Suc__0__iff, fact_nat__mult__commute, arity_Polynomial__Opoly__Divides_Oring__div, fact_mod__Suc__eq__Suc__mod, fact_mod__poly__eq, fact_add1__zle__eq, arity_Polynomial__Opoly__Rings_Olinordered__semiring, fact_eq__imp__le, fact_diff__add__inverse2, arity_Int__Oint__Rings_Olinordered__semiring__1, fact_diff__eq__diff__eq, fact_minus__le__self__iff, fact_le__less__Suc__eq, fact_not__pos__poly__0, fact_neq0__conv, fact_Suc__n__not__n, fact_minus__equation__iff, fact_mult__left__mono__neg, fact_mult__0__right, fact_le__antisym, fact_zless__linear, fact_trans__le__add1, fact_zadd__0__right, arity_Int__Oint__Rings_Olinordered__comm__semiring__strict, fact_diff__le__mono, arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring, fact_mult__Suc__right, arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le, arity_Polynomial__Opoly__Rings_Ocomm__ring__1, fact_synthetic__div__unique__lemma, fact_zdvd__period, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J, fact_degree__monom__le, arity_Polynomial__Opoly__Rings_Odvd, arity_Polynomial__Opoly__Rings_Ocomm__semiring, arity_Int__Oint__Groups_Ominus, fact_zmult__1__right, arity_Polynomial__Opoly__Rings_Ono__zero__divisors, fact_double__eq__0__iff, fact_le__0__eq, fact_zdvd__imp__le, fact_dvd__0__left, fact_pcompose__pCons, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J, fact_mult__less__cancel2, fact_zle__add1__eq__le, fact_dvd__1__iff__1, fact_mod__0, fact_mult__1__right, fact_zero__less__two, fact_gcd__lcm__complete__lattice__nat_Obot__least, fact_diff__is__0__eq, arity_Polynomial__Opoly__Groups_Ozero, fact_dvd__iff__poly__eq__0, arity_Nat__Onat__Rings_Osemiring, fact_termination__basic__simps_I1_J, fact_less__iff__Suc__add, fact_add__le__imp__le__right, fact_pdivmod__rel__unique__mod, clrel_Rings_Ocomm__semiring__0__Rings_Omult__zero, conj_0, fact_le__add__diff__inverse, fact_nat__zero__less__power__iff, fact_mod__add__right__eq, arity_Int__Oint__Rings_Ocomm__ring__1, fact_add__le__mono, fact_less__Suc0, fact_pos__poly__def, fact_dvd_Ole__less__trans, fact_diff__monom, fact_add__diff__inverse, fact_convex__bound__le, arity_Int__Oint__Groups_Olinordered__ab__group__add, fact_nat__mult__eq__cancel__disj, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J, fact_le__diff__iff, fact_smult__minus__right, arity_Int__Oint__Rings_Ozero__neq__one, fact_add__imp__eq, fact_power__gt1, fact_no__zero__divisors, fact_le__square, fact_zadd__zmult__distrib, fact_divisors__zero, arity_Polynomial__Opoly__Rings_Olinordered__ring, arity_Polynomial__Opoly__Groups_Ocomm__monoid__add, fact_degree__mult__eq, fact_ord__eq__le__trans, fact_zadd__zminus__inverse2, fact_double__add__less__zero__iff__single__add__less__zero, 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, fact_nat__less__le, fact_less__SucE, fact_le__cube, fact_zless__add1__eq, fact_add__less__mono1, fact_add__le__cancel__left, fact_smult__diff__left, arity_Polynomial__Opoly__Groups_Oone, fact_poly__eq__iff, fact_field__inverse__zero, fact_realpow__Suc__le__self, fact_one__le__inverse__iff, fact_zmod__zminus1__not__zero, fact_less__diff__iff, fact_pdivmod__rel__0, fact_odd__less__0, fact_dvd_Oless__imp__not__eq2, 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_power__le__imp__le__base, fact_mult__1, fact_pdivmod__rel__def, fact_mod__add__eq, fact_zdiv__mono2__neg__lemma, arity_Int__Oint__Divides_Oring__div, fact_dvd__imp__mod__0, fact_less__SucI, fact_Suc__not__Zero, fact_dvd__power, fact_poly__gcd_Osimps_I2_J, fact_not__square__less__zero, fact_Suc__lessI, fact_add__Suc__right, fact_mult__pCons__left, fact_mod__mult__distrib, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J, fact_coeff__diff, fact_dvd__mod__imp__dvd
% 32.50/14.39  % No. of inferences in proof: 2014
% 32.56/14.40  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 13866 ms resp. 12589 ms w/o parsing
% 36.65/15.77  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 36.93/15.82  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------