TSTP Solution File: SWW205+1 by Leo-III---1.7.15

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.15
% Problem  : SWW205+1 : TPTP v8.2.0. Released v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d THM

% Computer : n027.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Mon Jun 24 18:21:17 EDT 2024

% Result   : Theorem 55.21s 13.13s
% Output   : Refutation 56.58s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      : 1265
% Syntax   : Number of formulae    : 2532 ( 812 unt;   0 typ;   0 def)
%            Number of atoms       : 6382 (1659 equ;   0 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 35989 ( 263   ~; 132   |; 521   &;31876   @)
%                                         ( 231 <=>;2966  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   7 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  126 ( 124 usr;  12 con; 0-5 aty)
%            Number of variables   : 5486 (   0   ^5391   !;  95   ?;5486   :)

% Comments : 
%------------------------------------------------------------------------------
thf(c_SEQ_OBseq_type,type,
    c_SEQ_OBseq: $i > $i > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(class_Rings_Olinordered__semiring__1__strict_type,type,
    class_Rings_Olinordered__semiring__1__strict: $i > $o ).

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

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

thf(class_Rings_Olinordered__semiring__1_type,type,
    class_Rings_Olinordered__semiring__1: $i > $o ).

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

thf(class_Rings_Ocomm__semiring__0_type,type,
    class_Rings_Ocomm__semiring__0: $i > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(class_Lattices_Oboolean__algebra_type,type,
    class_Lattices_Oboolean__algebra: $i > $o ).

thf(c_Groups_Ouminus__class_Ouminus_type,type,
    c_Groups_Ouminus__class_Ouminus: $i > $i ).

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

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

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

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

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

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

thf(class_Rings_Osemiring__0_type,type,
    class_Rings_Osemiring__0: $i > $o ).

thf(c_COMBS_type,type,
    c_COMBS: $i > $i > $i > $i > $i > $i ).

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

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

thf(class_Rings_Oring__1_type,type,
    class_Rings_Oring__1: $i > $o ).

thf(class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_type,type,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct: $i > $o ).

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

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

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

thf(class_Rings_Oring__1__no__zero__divisors_type,type,
    class_Rings_Oring__1__no__zero__divisors: $i > $o ).

thf(class_Rings_Ocomm__semiring__1_type,type,
    class_Rings_Ocomm__semiring__1: $i > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(class_Groups_Oabs__if_type,type,
    class_Groups_Oabs__if: $i > $o ).

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

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

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

thf(class_Rings_Ocomm__ring__1_type,type,
    class_Rings_Ocomm__ring__1: $i > $o ).

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

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

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

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

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

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

thf(c_RealVector_Oof__real_type,type,
    c_RealVector_Oof__real: $i > $i ).

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

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

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

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

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

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

thf(class_RealVector_Oreal__normed__algebra__1_type,type,
    class_RealVector_Oreal__normed__algebra__1: $i > $o ).

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

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

thf(c_SEQ_OCauchy_type,type,
    c_SEQ_OCauchy: $i > $i > $o ).

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

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

thf(c_Lim_OisUCont_type,type,
    c_Lim_OisUCont: $i > $i > $i > $o ).

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

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

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

thf(class_RealVector_Oreal__algebra__1_type,type,
    class_RealVector_Oreal__algebra__1: $i > $o ).

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

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

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

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

thf(c_SEQ_Oconvergent_type,type,
    c_SEQ_Oconvergent: $i > $i > $o ).

thf(class_SEQ_Ocomplete__space_type,type,
    class_SEQ_Ocomplete__space: $i > $o ).

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

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

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

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

thf(class_RealVector_Ometric__space_type,type,
    class_RealVector_Ometric__space: $i > $o ).

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

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

thf(c_SEQ_Olim_type,type,
    c_SEQ_Olim: $i > $i > $i ).

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

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

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

thf(895,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Osgn__class_Osgn @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Osgn__class_Osgn @ C @ B ) ) @ ( c_Groups_Osgn__class_Osgn @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__times) ).

thf(4188,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Osgn__class_Osgn @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Osgn__class_Osgn @ C @ B ) ) @ ( c_Groups_Osgn__class_Osgn @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(810,axiom,
    c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J) ).

thf(3883,plain,
    c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(1152,axiom,
    class_Groups_Oordered__ab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add) ).

thf(4835,plain,
    class_Groups_Oordered__ab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).

thf(493,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) @ ( c_Transcendental_Oln @ A ) )
        <=> ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__less__cancel__iff) ).

thf(2778,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) @ ( c_Transcendental_Oln @ A ) )
           => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
           => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) @ ( c_Transcendental_Oln @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(1192,axiom,
    class_SEQ_Ocomplete__space @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__SEQ_Ocomplete__space) ).

thf(4899,plain,
    class_SEQ_Ocomplete__space @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).

thf(1164,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__Im__mult) ).

thf(4852,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1164]) ).

thf(409,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
        & ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__two__squares__add__zero__iff) ).

thf(2480,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(430,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__comm__semiring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict) ).

thf(2559,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__comm__semiring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(825,axiom,
    ( ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal )
   != ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__zero__not__eq__one) ).

thf(3927,plain,
    ( ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal )
   != ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(1066,axiom,
    class_Groups_Oab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__add) ).

thf(4682,plain,
    class_Groups_Oab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(567,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oab__group__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__left__minus) ).

thf(3021,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oab__group__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(32,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__mult) ).

thf(1322,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(167,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__power__nat__eq) ).

thf(1665,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) )
      & ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Oadd) ).

thf(1501,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(400,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring__1__no__zero__divisors @ C )
     => ( ( B
         != ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A )
         != ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_field__power__not__zero) ).

thf(2452,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring__1__no__zero__divisors @ C )
     => ( ( B
         != ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A )
         != ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(1008,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__minus__cancel) ).

thf(4560,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).

thf(835,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__equal__0__iff__equal) ).

thf(3962,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ~ ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
         => ~ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__leE) ).

thf(1879,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ~ ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
         => ~ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(40,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Groups_Osgn__class_Osgn @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
        = ( c_Groups_Osgn__class_Osgn @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__sgn) ).

thf(1337,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Groups_Osgn__class_Osgn @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
        = ( c_Groups_Osgn__class_Osgn @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(290,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__0__less__add__iff) ).

thf(2094,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(85,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zminus__zminus) ).

thf(1411,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(355,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) )
        = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
     => ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ B )
        = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__zmult__eq__1) ).

thf(2306,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) )
        = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
     => ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ B )
        = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(663,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Deriv_Oadd__diff__add) ).

thf(3345,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(984,axiom,
    ( ( c_Complex_Oexpi @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_expi__zero) ).

thf(4464,plain,
    ( ( c_Complex_Oexpi @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(367,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i,D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ A ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ A @ C ) ) @ ( c_RealVector_Onorm__class_Onorm @ A @ D ) ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Opos__bounded) ).

thf(2341,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i,D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ A ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ A @ C ) ) @ ( c_RealVector_Onorm__class_Onorm @ A @ D ) ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(1013,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
      <=> ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
          & ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__less__iff) ).

thf(4571,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
         => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
            & ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
            & ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).

thf(579,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_RComplete_Onatceiling @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__subtract) ).

thf(3053,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_RComplete_Onatceiling @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(562,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__commute) ).

thf(3008,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(298,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ ( c_Groups_Oone__class_Oone @ D ) )
           => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__decreasing) ).

thf(2131,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ ( c_Groups_Oone__class_Oone @ D ) )
           => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(1198,axiom,
    class_Rings_Ono__zero__divisors @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ono__zero__divisors) ).

thf(4909,plain,
    class_Rings_Ono__zero__divisors @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).

thf(922,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).

thf(4262,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(744,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Odiff) ).

thf(3661,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(192,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hAPP @ B @ C )
          = ( hAPP @ A @ C ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ext) ).

thf(1767,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hAPP @ B @ C )
          = ( hAPP @ A @ C ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(262,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ C )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__mono) ).

thf(1995,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ C )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(826,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Ominus) ).

thf(3930,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(746,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__i__mult__minus) ).

thf(3666,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(1092,axiom,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex )
    = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__zero__def) ).

thf(4729,plain,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex )
    = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).

thf(677,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oone @ B )
     => ( ( ( c_Groups_Oone__class_Oone @ B )
          = A )
      <=> ( A
          = ( c_Groups_Oone__class_Oone @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__reorient) ).

thf(3397,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oone @ B )
     => ( ( ( ( c_Groups_Oone__class_Oone @ B )
            = A )
         => ( A
            = ( c_Groups_Oone__class_Oone @ B ) ) )
        & ( ( A
            = ( c_Groups_Oone__class_Oone @ B ) )
         => ( ( c_Groups_Oone__class_Oone @ B )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(236,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).

thf(1899,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(439,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__strict__left__mono) ).

thf(2595,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(219,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ C )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less__mono) ).

thf(1850,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ C )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(720,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_RComplete_Onatfloor @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__subtract) ).

thf(3566,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_RComplete_Onatfloor @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(1091,axiom,
    class_Rings_Olinordered__comm__semiring__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__comm__semiring__strict) ).

thf(4728,plain,
    class_Rings_Olinordered__comm__semiring__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).

thf(132,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Power_Opower @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc) ).

thf(1554,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Power_Opower @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(611,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__cancel) ).

thf(3172,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(172,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) ) @ c_Complex_Oii )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Complex__mult__i) ).

thf(1686,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) ) @ c_Complex_Oii )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(889,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_q__neg__lemma) ).

thf(4166,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

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

thf(2250,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(860,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__commute) ).

thf(4043,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(614,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
              = ( c_Groups_Ozero__class_Ozero @ C ) )
          <=> ( ( B
                = ( c_Groups_Ozero__class_Ozero @ C ) )
              & ( A
                = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__nonneg__eq__0__iff) ).

thf(3185,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
                = ( c_Groups_Ozero__class_Ozero @ C ) )
             => ( ( B
                  = ( c_Groups_Ozero__class_Ozero @ C ) )
                & ( A
                  = ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
            & ( ( ( B
                  = ( c_Groups_Ozero__class_Ozero @ C ) )
                & ( A
                  = ( c_Groups_Ozero__class_Ozero @ C ) ) )
             => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
                = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(244,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__add__strict) ).

thf(1924,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

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

thf(1406,plain,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(383,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__le__zero__cancel__iff) ).

thf(2395,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(150,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Transcendental_Oln @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ ( c_Transcendental_Oln @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__realpow) ).

thf(1618,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Transcendental_Oln @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ ( c_Transcendental_Oln @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(670,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__le__D1) ).

thf(3377,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(180,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( ( B != A )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__cases) ).

thf(1728,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( ( B != A )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

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

thf(1269,plain,
    class_Groups_Ouminus @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(1029,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ v_s @ A ) ) @ v_r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_assms) ).

thf(4614,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ v_s @ A ) ) @ v_r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).

thf(1084,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( c_RComplete_Onatfloor @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__neg) ).

thf(4714,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( c_RComplete_Onatfloor @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).

thf(857,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__one__le__power) ).

thf(4037,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(356,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zmult__distrib) ).

thf(2309,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(15,axiom,
    class_Rings_Olinordered__ring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring) ).

thf(1290,plain,
    class_Rings_Olinordered__ring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(230,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( c_Nat_OSuc @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ ( c_Nat_OSuc @ B ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__le__imp__le__base) ).

thf(1882,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( c_Nat_OSuc @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ ( c_Nat_OSuc @ B ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(773,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ A )
     => ( ( c_RealVector_Onorm__class_Onorm @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__zero) ).

thf(3762,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ A )
     => ( ( c_RealVector_Onorm__class_Onorm @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(453,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__leI) ).

thf(2644,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(789,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__imp__le__right) ).

thf(3813,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(379,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I10_J) ).

thf(2381,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(1248,axiom,
    class_Groups_Oab__group__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__group__add) ).

thf(5000,plain,
    class_Groups_Oab__group__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1248]) ).

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

thf(1808,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
       => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(683,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__ring__abs @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__ring__abs) ).

thf(3417,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__ring__abs @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(437,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ ( hAPP @ v_s @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ v_r ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_th) ).

thf(2590,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ ( hAPP @ v_s @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ v_r ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(78,axiom,
    class_Rings_Ocomm__ring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring) ).

thf(1401,plain,
    class_Rings_Ocomm__ring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(92,axiom,
    class_Rings_Olinordered__ring__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring__strict) ).

thf(1422,plain,
    class_Rings_Olinordered__ring__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

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

thf(1569,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) ) @ A )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(338,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Ocomm__semiring__0 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__0) ).

thf(2248,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Ocomm__semiring__0 @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(1137,axiom,
    class_Rings_Oring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring) ).

thf(4807,plain,
    class_Rings_Oring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).

thf(820,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
        & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel1) ).

thf(3912,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
          & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
          & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(590,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
      <=> ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__cancel__right) ).

thf(3098,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ D @ C @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(463,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ A )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        = ( c_Groups_Oone__class_Oone @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__1) ).

thf(2676,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ A )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        = ( c_Groups_Oone__class_Oone @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(819,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__mono) ).

thf(3910,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(1083,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ B )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__antisym) ).

thf(4711,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ B )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).

thf(1024,axiom,
    ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__Im__one) ).

thf(4603,plain,
    ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).

thf(291,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__commute) ).

thf(2099,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(141,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ B )
     => ( ( c_Groups_Osgn__class_Osgn @ B @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) )
        = ( hAPP @ ( c_RealVector_Oof__real @ B ) @ ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__of__real) ).

thf(1591,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ B )
     => ( ( c_Groups_Osgn__class_Osgn @ B @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) )
        = ( hAPP @ ( c_RealVector_Oof__real @ B ) @ ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(174,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ A @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__self) ).

thf(1692,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ A @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(1035,axiom,
    class_Orderings_Oord @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oord) ).

thf(4623,plain,
    class_Orderings_Oord @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).

thf(1231,axiom,
    class_Power_Opower @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Power_Opower) ).

thf(4973,plain,
    class_Power_Opower @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1231]) ).

thf(1102,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( c_RComplete_Onatfloor @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__natfloor) ).

thf(4747,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( c_RComplete_Onatfloor @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).

thf(98,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add2) ).

thf(1437,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(991,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos2) ).

thf(4487,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(1021,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I8_J) ).

thf(4592,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).

thf(910,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( c_Groups_Oone__class_Oone @ C ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) ).

thf(4234,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( c_Groups_Oone__class_Oone @ C ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(1040,axiom,
    class_Groups_Oone @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oone) ).

thf(4633,plain,
    class_Groups_Oone @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(1014,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__power__ineq) ).

thf(4575,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).

thf(659,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ A @ ( c_Groups_Oone__class_Oone @ A ) )
        = ( c_Groups_Oone__class_Oone @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__one) ).

thf(3332,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ A @ ( c_Groups_Oone__class_Oone @ A ) )
        = ( c_Groups_Oone__class_Oone @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(661,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_q__pos__lemma) ).

thf(3338,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(8,axiom,
    class_Rings_Olinordered__semiring__1__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1__strict) ).

thf(1275,plain,
    class_Rings_Olinordered__semiring__1__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(733,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__1) ).

thf(3616,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(633,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq2) ).

thf(3253,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(157,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( c_Orderings_Oord__class_Oless__eq @ A @ ( c_Groups_Oone__class_Oone @ A ) @ ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__one__le__zero) ).

thf(1640,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( c_Orderings_Oord__class_Oless__eq @ A @ ( c_Groups_Oone__class_Oone @ A ) @ ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(275,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__idom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oabs__class_Oabs @ D @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) @ A )
      <=> ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) @ C )
          & ( c_Orderings_Oord__class_Oless @ D @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__diff__less__iff) ).

thf(2042,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__idom @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oabs__class_Oabs @ D @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) @ A )
         => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) @ C )
            & ( c_Orderings_Oord__class_Oless @ D @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) @ C )
            & ( c_Orderings_Oord__class_Oless @ D @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) )
         => ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oabs__class_Oabs @ D @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(640,axiom,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Ononneg__bounded) ).

thf(3276,plain,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(854,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ C )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__strict__mono) ).

thf(4026,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ C )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(575,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__order) ).

thf(3044,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(183,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__le__cases) ).

thf(1735,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(224,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I5_J) ).

thf(1867,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(612,axiom,
    ! [A: $i,B: $i] :
      ( ( ( class_Power_Opower @ B )
        & ( class_Rings_Osemiring__0 @ B ) )
     => ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
            = ( c_Groups_Oone__class_Oone @ B ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__0__left) ).

thf(3175,plain,
    ! [A: $i,B: $i] :
      ( ( ( class_Power_Opower @ B )
        & ( class_Rings_Osemiring__0 @ B ) )
     => ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
            = ( c_Groups_Oone__class_Oone @ B ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(1166,axiom,
    ? [A: $i] :
    ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Obounded) ).

thf(4856,plain,
    ? [A: $i] :
    ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1166]) ).

thf(976,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) @ B )
        = ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__diff__le) ).

thf(4440,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) @ B )
        = ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(449,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__strict__right__mono__neg) ).

thf(2627,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(313,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__leE) ).

thf(2168,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(43,axiom,
    class_Rings_Olinordered__semiring__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__strict) ).

thf(1342,plain,
    class_Rings_Olinordered__semiring__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(597,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ A )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__inverse2) ).

thf(3121,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(932,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( ( c_RComplete_Onatfloor @ A )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__eq) ).

thf(4294,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( ( c_RComplete_Onatfloor @ A )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(532,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_times_Oidem) ).

thf(2901,plain,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(653,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel1) ).

thf(3313,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(50,axiom,
    class_Orderings_Oorder @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Oorder) ).

thf(1354,plain,
    class_Orderings_Oorder @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(303,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
        | ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__linear) ).

thf(2146,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
        | ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(814,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__1__right) ).

thf(3895,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(374,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_RealVector_Onorm__class_Onorm @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__eq__zero) ).

thf(2360,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( ( c_RealVector_Onorm__class_Onorm @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_RealVector_Onorm__class_Onorm @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(1111,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__comm__monoid__add) ).

thf(4762,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).

thf(104,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) @ ( c_Transcendental_Oln @ A ) )
        <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__le__cancel__iff) ).

thf(1458,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) @ ( c_Transcendental_Oln @ A ) )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) @ ( c_Transcendental_Oln @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(1214,axiom,
    c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ v_s @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ v_r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact__096cmod_A_Is_A0_J_A_060_061_Ar_096) ).

thf(4941,plain,
    c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ v_s @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ v_r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1214]) ).

thf(660,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__is__0__eq_H) ).

thf(3335,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(187,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
    <=> ? [B: $i] :
          ( A
          = ( c_Nat_OSuc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gr0__conv__Suc) ).

thf(1746,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ? [B: $i] :
            ( A
            = ( c_Nat_OSuc @ B ) ) )
      & ( ? [B: $i] :
            ( A
            = ( c_Nat_OSuc @ B ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(241,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__ge__self) ).

thf(1914,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(704,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Osemiring @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_combine__common__factor) ).

thf(3502,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Osemiring @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) ) @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(74,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) @ ( c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mono) ).

thf(1391,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) @ ( c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(870,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__imp__not__eq2) ).

thf(4079,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(586,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__left__mono) ).

thf(3088,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(1203,axiom,
    class_Orderings_Olinorder @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Olinorder) ).

thf(4916,plain,
    class_Orderings_Olinorder @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1203]) ).

thf(70,axiom,
    class_Rings_Olinordered__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring) ).

thf(1386,plain,
    class_Rings_Olinordered__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(815,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__greater) ).

thf(3898,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(856,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) )
      <=> ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
          | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__0__iff) ).

thf(4030,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) ) ) )
        & ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(315,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__le__cancel1) ).

thf(2173,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(419,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__minus) ).

thf(2516,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(1110,axiom,
    ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__Re__one) ).

thf(4760,plain,
    ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).

thf(941,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ F @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__iff2) ).

thf(4322,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ F @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ F @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(300,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ E )
     => ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Oabs__class_Oabs @ E @ ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( c_Groups_Oabs__class_Oabs @ E @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) ) @ ( c_Groups_Oabs__class_Oabs @ E @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__diff__triangle__ineq) ).

thf(2136,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ E )
     => ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Oabs__class_Oabs @ E @ ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( c_Groups_Oabs__class_Oabs @ E @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) ) @ ( c_Groups_Oabs__class_Oabs @ E @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(385,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
      <=> ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
          & ~ ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__le__not__le) ).

thf(2405,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ~ ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ~ ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(549,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I7_J) ).

thf(2957,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(797,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__cancel__right) ).

thf(3836,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(947,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring__abs @ C )
     => ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            | ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
            | ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
       => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__eq__mult) ).

thf(4343,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring__abs @ C )
     => ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            | ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
            | ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
       => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(370,axiom,
    ! [A: $i,B: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ B )
        & ( class_RealVector_Oreal__normed__vector @ B ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Ominus) ).

thf(2348,plain,
    ! [A: $i,B: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ B )
        & ( class_RealVector_Oreal__normed__vector @ B ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(716,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B )
          = ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__eq__iff) ).

thf(3545,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B )
            = ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B )
            = ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(276,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oabs__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oabs__if) ).

thf(2046,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oabs__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(554,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
      <=> ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
            & ( c_Orderings_Oord__class_Oless @ D @ B @ A ) )
          | ( ( c_Orderings_Oord__class_Oless @ D @ C @ ( c_Groups_Ozero__class_Ozero @ D ) )
            & ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel__left__disj) ).

thf(2972,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
         => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
              & ( c_Orderings_Oord__class_Oless @ D @ B @ A ) )
            | ( ( c_Orderings_Oord__class_Oless @ D @ C @ ( c_Groups_Ozero__class_Ozero @ D ) )
              & ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) )
        & ( ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
              & ( c_Orderings_Oord__class_Oless @ D @ B @ A ) )
            | ( ( c_Orderings_Oord__class_Oless @ D @ C @ ( c_Groups_Ozero__class_Ozero @ D ) )
              & ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(737,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_COMBC @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__COMBC__1) ).

thf(3634,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_COMBC @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(487,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__neq__le__trans) ).

thf(2763,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

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

thf(2548,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(197,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__mono1) ).

thf(1782,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(269,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
          = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat_Odiff__diff__eq) ).

thf(2020,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
          = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(708,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__diff__add__eq1) ).

thf(3513,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(1065,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ ( c_Complex_Ocomplex_OComplex @ D @ C ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ D @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__diff) ).

thf(4679,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ ( c_Complex_Ocomplex_OComplex @ D @ C ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ D @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(651,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__assoc) ).

thf(3308,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(248,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
           => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
             => ( C = A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__eq__imp__eq__base) ).

thf(1935,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
           => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
             => ( C = A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(1133,axiom,
    class_Orderings_Oorder @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Oorder) ).

thf(4803,plain,
    class_Orderings_Oorder @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).

thf(763,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_compl__mono) ).

thf(3730,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(159,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ono__zero__divisors @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ C ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_divisors__zero) ).

thf(1645,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ono__zero__divisors @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ C ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(152,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Oab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__group__add) ).

thf(1623,plain,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Oab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(111,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Oadd__right) ).

thf(1482,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(998,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zminus__zadd__distrib) ).

thf(4524,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(829,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) )
      = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__minus__add__cancel) ).

thf(3938,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) )
      = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(1009,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__gt1) ).

thf(4563,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).

thf(274,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zpower__zpower) ).

thf(2039,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(968,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less__mono2) ).

thf(4415,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(238,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Ocomm__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring) ).

thf(1907,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Ocomm__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(1054,axiom,
    class_Rings_Oring__1__no__zero__divisors @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors) ).

thf(4654,plain,
    class_Rings_Oring__1__no__zero__divisors @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(700,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ozero__neq__one @ A )
     => ( ( c_Groups_Oone__class_Oone @ A )
       != ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__neq__zero) ).

thf(3485,plain,
    ! [A: $i] :
      ( ( class_Rings_Ozero__neq__one @ A )
     => ( ( c_Groups_Oone__class_Oone @ A )
       != ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(1114,axiom,
    class_Groups_Oab__semigroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__add) ).

thf(4766,plain,
    class_Groups_Oab__semigroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).

thf(480,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__cube) ).

thf(2745,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(914,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__natfloor__add__one__gt) ).

thf(4243,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(631,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__abs) ).

thf(3248,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(84,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__trans) ).

thf(1409,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(635,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__int__def__symmetric) ).

thf(3263,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(622,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__le__imp__le) ).

thf(3221,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(1170,axiom,
    class_Rings_Oring__1__no__zero__divisors @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__1__no__zero__divisors) ).

thf(4866,plain,
    class_Rings_Oring__1__no__zero__divisors @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).

thf(1189,axiom,
    class_Groups_Oone @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oone) ).

thf(4896,plain,
    class_Groups_Oone @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).

thf(20,axiom,
    class_Rings_Olinordered__semidom @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semidom) ).

thf(1298,plain,
    class_Rings_Olinordered__semidom @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(189,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less) ).

thf(1760,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(600,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( c_Groups_Oone__class_Oone @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) ).

thf(3131,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( c_Groups_Oone__class_Oone @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(595,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( c_SEQ_Oconvergent @ C @ ( c_COMBC @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Oconvergent) ).

thf(3116,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( c_SEQ_Oconvergent @ C @ ( c_COMBC @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(459,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__mono) ).

thf(2667,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(366,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( ( c_SEQ_Oconvergent @ C @ A )
         => ( c_SEQ_Oconvergent @ C @ ( c_COMBS @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Oconvergent) ).

thf(2339,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( ( c_SEQ_Oconvergent @ C @ A )
         => ( c_SEQ_Oconvergent @ C @ ( c_COMBS @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(576,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__imp__neg__le) ).

thf(3046,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(752,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
    <=> ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__le__real__less) ).

thf(3689,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(906,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__minus__iff) ).

thf(4224,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(226,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__minus__commute) ).

thf(1871,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(470,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ C )
     => ( ( B
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0__iff) ).

thf(2708,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ C )
     => ( ( ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ C ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(792,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
       => ? [D: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ D )
            & ! [E: $i] :
                ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ E ) ) @ D )
               => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C @ E )
                  & ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ E @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_lemma__interval__lt) ).

thf(3821,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
       => ? [D: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ D )
            & ! [E: $i] :
                ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ E ) ) @ D )
               => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C @ E )
                  & ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ E @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(33,axiom,
    c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).

thf(1323,plain,
    c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(742,axiom,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ? [B: $i] :
        ! [C: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( c_RealVector_Oof__real @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Obounded) ).

thf(3653,plain,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ? [B: $i] :
        ! [C: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( c_RealVector_Oof__real @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ C ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(1124,axiom,
    ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__zero) ).

thf(4778,plain,
    ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).

thf(775,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__diff) ).

thf(3776,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(465,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__add) ).

thf(2682,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(213,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
      <=> ( ( B
            = ( c_Groups_Ozero__class_Ozero @ C ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sum__squares__eq__zero__iff) ).

thf(1826,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
        & ( ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(25,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__commute) ).

thf(1313,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(11,axiom,
    class_Rings_Omult__zero @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Omult__zero) ).

thf(1279,plain,
    class_Rings_Omult__zero @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(927,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) ).

thf(4275,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(318,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_leD) ).

thf(2185,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(138,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ C ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_left__add__mult__distrib) ).

thf(1577,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ C ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(129,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ ( c_COMBK @ E @ F @ C ) @ B ) @ A )
      = C ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_K__record__comp) ).

thf(1545,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ ( c_COMBK @ E @ F @ C ) @ B ) @ A )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(243,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__le__add1) ).

thf(1922,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(516,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le0) ).

thf(2857,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(1031,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__neqE__nat) ).

thf(4617,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).

thf(897,axiom,
    ! [A: $i,B: $i] :
      ( ( class_SEQ_Ocomplete__space @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
      <=> ( c_SEQ_Oconvergent @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Cauchy__convergent__iff) ).

thf(4194,plain,
    ! [A: $i,B: $i] :
      ( ( class_SEQ_Ocomplete__space @ B )
     => ( ( ( c_SEQ_OCauchy @ B @ A )
         => ( c_SEQ_Oconvergent @ B @ A ) )
        & ( ( c_SEQ_Oconvergent @ B @ A )
         => ( c_SEQ_OCauchy @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(29,axiom,
    class_Groups_Oordered__ab__group__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__group__add) ).

thf(1319,plain,
    class_Groups_Oordered__ab__group__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(1059,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Complex_Ocomplex_OComplex @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
        & ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Complex__eq__0) ).

thf(4664,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Complex_Ocomplex_OComplex @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
       => ( ( c_Complex_Ocomplex_OComplex @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(925,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__pos__pos) ).

thf(4270,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(1150,axiom,
    ( ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex )
    = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__one__def) ).

thf(4830,plain,
    ( ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex )
    = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).

thf(695,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diffs0__imp__equal) ).

thf(3470,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(1222,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
      | ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__le__linear) ).

thf(4952,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
      | ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1222]) ).

thf(832,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__pos__neg2) ).

thf(3951,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(923,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__increasing2) ).

thf(4265,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(208,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__minus__mult__self__le) ).

thf(1815,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(718,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
      <=> ( A
         != ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__abs__iff) ).

thf(3557,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
         => ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(65,axiom,
    class_Rings_Ono__zero__divisors @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ono__zero__divisors) ).

thf(1379,plain,
    class_Rings_Ono__zero__divisors @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(669,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Fun_Ocomp @ G @ F @ E @ D @ ( c_Fun_Ocomp @ C @ G @ E @ B @ A ) )
      = ( c_Fun_Ocomp @ C @ F @ E @ ( c_Fun_Ocomp @ G @ F @ C @ D @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_o__assoc) ).

thf(3374,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Fun_Ocomp @ G @ F @ E @ D @ ( c_Fun_Ocomp @ C @ G @ E @ B @ A ) )
      = ( c_Fun_Ocomp @ C @ F @ E @ ( c_Fun_Ocomp @ G @ F @ C @ D @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(1034,axiom,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) ).

thf(4622,plain,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).

thf(47,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_COMBS @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__COMBS__1) ).

thf(1347,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_COMBS @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(137,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__add1__eq__le) ).

thf(1572,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(960,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__le__imp__less__or__eq) ).

thf(4390,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
          | ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(987,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__le__mono) ).

thf(4471,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(1119,axiom,
    class_Groups_Oordered__ab__group__add__abs @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs) ).

thf(4771,plain,
    class_Groups_Oordered__ab__group__add__abs @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).

thf(626,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat_Oadd__0__right) ).

thf(3231,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(1227,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__1) ).

thf(4967,plain,
    class_Rings_Ocomm__semiring__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1227]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) )
      <=> ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__less__iff1) ).

thf(3268,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(1108,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__Re__mult) ).

thf(4756,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).

thf(506,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add) ).

thf(2829,plain,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(779,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__mod__triangle__ineq2) ).

thf(3788,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(330,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__mono__neg) ).

thf(2222,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(655,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).

thf(3322,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(812,axiom,
    ! [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
    <=> ! [B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) @ ( hAPP @ A @ ( c_Nat_OSuc @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_subseq__Suc__iff) ).

thf(3887,plain,
    ! [A: $i] :
      ( ( ( c_SEQ_Osubseq @ A )
       => ! [B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) @ ( hAPP @ A @ ( c_Nat_OSuc @ B ) ) ) )
      & ( ! [B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) @ ( hAPP @ A @ ( c_Nat_OSuc @ B ) ) )
       => ( c_SEQ_Osubseq @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(933,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
    <=> ( ( C = A )
        | ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__cancel2) ).

thf(4297,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( ( C = A )
          | ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( C = A )
          | ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(90,axiom,
    class_Groups_Ominus @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ominus) ).

thf(1418,plain,
    class_Groups_Ominus @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(1159,axiom,
    class_Rings_Ono__zero__divisors @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ono__zero__divisors) ).

thf(4847,plain,
    class_Rings_Ono__zero__divisors @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).

thf(735,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__antisym__conv2) ).

thf(3621,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
           => ( B = A ) )
          & ( ( B = A )
           => ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(68,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__mult) ).

thf(1383,plain,
    class_Groups_Oab__semigroup__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(1070,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ A )
     => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__Cauchy__convergent) ).

thf(4686,plain,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ A )
     => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).

thf(1139,axiom,
    class_Groups_Ogroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ogroup__add) ).

thf(4810,plain,
    class_Groups_Ogroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).

thf(268,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__diff__eq) ).

thf(2017,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(973,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__triangle__ineq4) ).

thf(4432,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(264,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__natfloor__gt__diff__one) ).

thf(2007,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(546,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Ono__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ono__zero__divisors) ).

thf(2950,plain,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Ono__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(1073,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_OCauchy) ).

thf(4696,plain,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(901,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__eq__0__iff) ).

thf(4205,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

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

thf(1415,plain,
    class_Groups_Oordered__ab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(445,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ A )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__cancel) ).

thf(2616,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(962,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__div__algebra @ C )
     => ( ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__mult) ).

thf(4396,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__div__algebra @ C )
     => ( ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

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

thf(1465,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(22,axiom,
    class_Groups_Omonoid__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__mult) ).

thf(1302,plain,
    class_Groups_Omonoid__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(497,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) ).

thf(2794,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(1196,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OBseq @ tc_RealDef_Oreal @ A )
     => ( ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ A )
       => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__monoseq__convergent) ).

thf(4903,plain,
    ! [A: $i] :
      ( ( c_SEQ_OBseq @ tc_RealDef_Oreal @ A )
     => ( ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ A )
       => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1196]) ).

thf(977,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ B )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) ).

thf(4443,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ B )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(293,axiom,
    ! [A: $i] :
      ( ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
      & ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
          & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__sgn__def) ).

thf(2105,plain,
    ! [A: $i] :
      ( ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
      & ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
          & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(102,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) ).

thf(1450,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(288,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Oone__class_Oone @ A ) @ ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__one__less__zero) ).

thf(2090,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Oone__class_Oone @ A ) @ ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(1262,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_Oconvergent @ B @ A )
      <=> ( c_SEQ_Oconvergent @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convergent__minus__iff) ).

thf(5027,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_Oconvergent @ B @ A )
         => ( c_SEQ_Oconvergent @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
        & ( ( c_SEQ_Oconvergent @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( c_SEQ_Oconvergent @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1262]) ).

thf(675,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__neg__neg) ).

thf(3393,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(62,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( class_Orderings_Opreorder @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Orderings_Opreorder) ).

thf(1373,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( class_Orderings_Opreorder @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(560,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A )
      <=> ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__le__iff) ).

thf(2996,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ B ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(981,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__0) ).

thf(4454,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(647,axiom,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Ononneg__bounded) ).

thf(3295,plain,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(343,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
    <=> ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__less__le) ).

thf(2259,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
          & ( B != A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
          & ( B != A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(847,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__of__nonneg) ).

thf(4001,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(1144,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__0) ).

thf(4820,plain,
    class_Rings_Ocomm__semiring__0 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1144]) ).

thf(608,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__le__cancel__iff1) ).

thf(3163,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(696,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__minus__self__iff) ).

thf(3473,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(841,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__ab__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__imp__eq) ).

thf(3983,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__ab__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(689,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ominus @ E )
     => ( ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ E ) @ C @ B ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_fun__diff__def) ).

thf(3438,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ominus @ E )
     => ( ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ E ) @ C @ B ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(886,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C )
            = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__diff__iff) ).

thf(4147,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C )
              = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C )
              = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(223,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__diff__cancel) ).

thf(1864,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(296,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
      <=> ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) )
          | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__mult__iff) ).

thf(2122,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
         => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) )
            | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) )
        & ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) )
            | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
              & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(721,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ozero @ B )
     => ( ( ( c_Groups_Ozero__class_Ozero @ B )
          = A )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__reorient) ).

thf(3569,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ozero @ B )
     => ( ( ( ( c_Groups_Ozero__class_Ozero @ B )
            = A )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Ozero__class_Ozero @ B )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(169,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__mult__pos2) ).

thf(1677,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(425,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
      <=> ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_pos__zmult__eq__1__iff) ).

thf(2540,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
         => ( ( B
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
            & ( A
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
        & ( ( ( B
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
            & ( A
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(1173,axiom,
    class_Rings_Omult__zero @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Omult__zero) ).

thf(4871,plain,
    class_Rings_Omult__zero @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).

thf(645,axiom,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__real__of__nat__less__zero) ).

thf(3290,plain,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(588,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ozero__neq__one @ A )
     => ( ( c_Groups_Ozero__class_Ozero @ A )
       != ( c_Groups_Oone__class_Oone @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__neq__one) ).

thf(3093,plain,
    ! [A: $i] :
      ( ( class_Rings_Ozero__neq__one @ A )
     => ( ( c_Groups_Ozero__class_Ozero @ A )
       != ( c_Groups_Oone__class_Oone @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(842,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Ocomm__monoid__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult) ).

thf(3986,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Ocomm__monoid__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(1086,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Ometric__space @ B )
     => ( c_SEQ_Oconvergent @ B @ ( c_COMBK @ B @ tc_Nat_Onat @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convergent__const) ).

thf(4718,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Ometric__space @ B )
     => ( c_SEQ_Oconvergent @ B @ ( c_COMBK @ B @ tc_Nat_Onat @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).

thf(508,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).

thf(2833,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(1157,axiom,
    ( ( c_RComplete_Onatfloor @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__zero) ).

thf(4842,plain,
    ( ( c_RComplete_Onatfloor @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1157]) ).

thf(685,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
        & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel2) ).

thf(3421,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(759,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__diff__def) ).

thf(3712,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(101,axiom,
    ! [A: $i] :
      ( ( class_Power_Opower @ A )
     => ( ( c_Power_Opower__class_Opower @ A )
        = ( c_Power_Opower_Opower @ A @ ( c_Groups_Oone__class_Oone @ A ) @ ( c_Groups_Otimes__class_Otimes @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__power__power) ).

thf(1447,plain,
    ! [A: $i] :
      ( ( class_Power_Opower @ A )
     => ( ( c_Power_Opower__class_Opower @ A )
        = ( c_Power_Opower_Opower @ A @ ( c_Groups_Oone__class_Oone @ A ) @ ( c_Groups_Otimes__class_Otimes @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(787,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__double__add__iff__zero__le__single__add) ).

thf(3808,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(934,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semidom) ).

thf(4307,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(49,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oboolean__algebra @ B )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__compl) ).

thf(1351,plain,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oboolean__algebra @ B )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(1147,axiom,
    class_Rings_Osemiring__0 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Osemiring__0) ).

thf(4825,plain,
    class_Rings_Osemiring__0 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).

thf(1002,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) )
      <=> ( ( B
            = ( c_Groups_Ozero__class_Ozero @ C ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sum__squares__le__zero__iff) ).

thf(4536,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
        & ( ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(551,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__funD) ).

thf(2965,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(902,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__comm__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__mult__left__mono) ).

thf(4211,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__comm__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(863,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ E )
     => ( ( ( c_Groups_Ominus__class_Ominus @ E @ D @ C )
          = ( c_Groups_Ominus__class_Ominus @ E @ B @ A ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
        <=> ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__eq__diff__less__eq) ).

thf(4055,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ E )
     => ( ( ( c_Groups_Ominus__class_Ominus @ E @ D @ C )
          = ( c_Groups_Ominus__class_Ominus @ E @ B @ A ) )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
           => ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
           => ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(908,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__mono2) ).

thf(4230,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(1211,axiom,
    ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Ozero) ).

thf(4935,plain,
    ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1211]) ).

thf(705,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) )
        = ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__minus__commute) ).

thf(3505,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) )
        = ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(232,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_n__less__n__mult__m) ).

thf(1886,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

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

thf(2424,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(555,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Oab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__add) ).

thf(2979,plain,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Oab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(76,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__0__0) ).

thf(1394,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(1246,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( c_RComplete_Onatceiling @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__neg) ).

thf(4996,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( c_RComplete_Onatceiling @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1246]) ).

thf(336,axiom,
    ! [A: $i,B: $i] :
      ( ( B
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        | ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__self__implies__10) ).

thf(2242,plain,
    ! [A: $i,B: $i] :
      ( ( B
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        | ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(46,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oboolean__algebra @ B )
     => ( class_Lattices_Oboolean__algebra @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Lattices_Oboolean__algebra) ).

thf(1345,plain,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oboolean__algebra @ B )
     => ( class_Lattices_Oboolean__algebra @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(97,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
      <=> ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
          & ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__le__iff) ).

thf(1433,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(1077,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_Oconvergent @ B @ A )
       => ( ( c_SEQ_Olim @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_SEQ_Olim @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_lim__uminus) ).

thf(4701,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_Oconvergent @ B @ A )
       => ( ( c_SEQ_Olim @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_SEQ_Olim @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).

thf(482,axiom,
    ( c_Complex_Oii
   != ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__i__not__one) ).

thf(2750,plain,
    ( c_Complex_Oii
   != ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(80,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) ).

thf(1405,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(624,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I2_J) ).

thf(3226,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(1046,axiom,
    class_Groups_Oordered__comm__monoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add) ).

thf(4643,plain,
    class_Groups_Oordered__comm__monoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).

thf(251,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( c_Lim_OisUCont @ B @ B @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_OisUCont) ).

thf(1943,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( c_Lim_OisUCont @ B @ B @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(724,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
        <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__diff__iff) ).

thf(3588,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(956,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__left__mono) ).

thf(4379,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(1185,axiom,
    class_Rings_Oordered__ring__abs @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__ring__abs) ).

thf(4892,plain,
    class_Rings_Oordered__ring__abs @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).

thf(176,axiom,
    ! [A: $i] :
      ( ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
      & ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
          & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__real__def) ).

thf(1697,plain,
    ! [A: $i] :
      ( ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
      & ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
          & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(619,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__less__power) ).

thf(3211,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(124,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_leI) ).

thf(1521,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(146,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__0__less__iff__less) ).

thf(1604,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(563,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__ring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__ring__strict) ).

thf(3011,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__ring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(1182,axiom,
    class_Rings_Ozero__neq__one @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ozero__neq__one) ).

thf(4888,plain,
    class_Rings_Ozero__neq__one @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).

thf(1255,axiom,
    class_Groups_Oab__semigroup__mult @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__semigroup__mult) ).

thf(5017,plain,
    class_Groups_Oab__semigroup__mult @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1255]) ).

thf(311,axiom,
    ( ( hAPP @ c_Complex_OIm @ c_Complex_Oii )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__Im__i) ).

thf(2164,plain,
    ( ( hAPP @ c_Complex_OIm @ c_Complex_Oii )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(177,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Opreorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Opreorder) ).

thf(1708,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Opreorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(572,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__triangle__ineq) ).

thf(3037,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(1076,axiom,
    class_Rings_Olinordered__semiring__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__strict) ).

thf(4700,plain,
    class_Rings_Olinordered__semiring__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(161,axiom,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__nat__zero__code) ).

thf(1651,plain,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(1253,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__not__refl3) ).

thf(5012,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1253]) ).

thf(692,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Complex_Ocomplex_OComplex @ D @ C )
        = ( c_Complex_Ocomplex_OComplex @ B @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex_Oinject) ).

thf(3450,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_Complex_Ocomplex_OComplex @ D @ C )
          = ( c_Complex_Ocomplex_OComplex @ B @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( c_Complex_Ocomplex_OComplex @ D @ C )
          = ( c_Complex_Ocomplex_OComplex @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(904,axiom,
    ! [A: $i] :
      ( ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ A )
      = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__norm__def) ).

thf(4216,plain,
    ! [A: $i] :
      ( ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ A )
      = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(1186,axiom,
    class_Rings_Oordered__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__semiring) ).

thf(4893,plain,
    class_Rings_Oordered__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).

thf(725,axiom,
    ! [A: $i,B: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__add__less2) ).

thf(3591,plain,
    ! [A: $i,B: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(535,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ C ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_decr__lemma) ).

thf(2910,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ C ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(649,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
      <=> ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__not__le) ).

thf(3301,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ~ ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) )
        & ( ( c_Orderings_Oord__class_Oless @ C @ A @ B )
         => ~ ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(324,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Oring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring) ).

thf(2204,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Oring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(1105,axiom,
    class_Rings_Omult__zero @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Omult__zero) ).

thf(4753,plain,
    class_Rings_Omult__zero @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).

thf(610,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__mult) ).

thf(3169,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(454,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__int__def) ).

thf(2646,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(869,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_COMBK @ D @ C @ B ) @ A )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__COMBK__1) ).

thf(4076,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_COMBK @ D @ C @ B ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(478,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__power) ).

thf(2737,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(452,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__1) ).

thf(2641,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(1199,axiom,
    ? [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
      & ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Nat_Onat @ tc_RealDef_Oreal @ tc_Nat_Onat @ ( c_Fun_Ocomp @ tc_Nat_Onat @ tc_RealDef_Oreal @ tc_Nat_Onat @ ( c_Fun_Ocomp @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ v_s ) @ v_f____ ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact__096EX_Afa_O_Asubseq_Afa_A_G_Amonoseq_A_I_Fn_O_A_IIm_Ao_As_Ao_Af_J_A_Ifa_An_J_J_096) ).

thf(4910,plain,
    ? [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
      & ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Nat_Onat @ tc_RealDef_Oreal @ tc_Nat_Onat @ ( c_Fun_Ocomp @ tc_Nat_Onat @ tc_RealDef_Oreal @ tc_Nat_Onat @ ( c_Fun_Ocomp @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ v_s ) @ v_f____ ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).

thf(285,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__cancel__left) ).

thf(2083,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(874,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
        <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__increasing__iff) ).

thf(4100,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
           => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(447,axiom,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Opos__bounded) ).

thf(2621,plain,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(495,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
        | ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__le__less__linear) ).

thf(2790,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
        | ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(279,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__le__less__trans) ).

thf(2061,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(491,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sum__squares__ge__zero) ).

thf(2773,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(1247,axiom,
    class_Orderings_Opreorder @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Opreorder) ).

thf(4999,plain,
    class_Orderings_Opreorder @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1247]) ).

thf(765,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__increasing) ).

thf(3735,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(71,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__semigroup__add) ).

thf(1387,plain,
    class_Groups_Ocancel__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(985,axiom,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( c_RealVector_Oof__real @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ C ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Opos__bounded) ).

thf(4466,plain,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( c_RealVector_Oof__real @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ C ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(729,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ ( c_RComplete_Onatfloor @ A ) ) @ ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__mult__natfloor) ).

thf(3603,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ ( c_RComplete_Onatfloor @ A ) ) @ ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(974,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Complex_Ocomplex_Ocomplex__rec @ D @ C @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex_Orecs) ).

thf(4434,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Complex_Ocomplex_Ocomplex__rec @ D @ C @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(918,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__mult__ineq) ).

thf(4252,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(731,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).

thf(3610,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(1145,axiom,
    class_Rings_Olinordered__semiring__strict @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring__strict) ).

thf(4821,plain,
    class_Rings_Olinordered__semiring__strict @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).

thf(852,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Opos__bounded) ).

thf(4021,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(1202,axiom,
    class_Rings_Ono__zero__divisors @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ono__zero__divisors) ).

thf(4915,plain,
    class_Rings_Ono__zero__divisors @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).

thf(386,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ B )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I1_J) ).

thf(2409,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ B )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(44,axiom,
    class_Groups_Omonoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__add) ).

thf(1343,plain,
    class_Groups_Omonoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(837,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__left__imp__eq) ).

thf(3972,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(1205,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__comm__monoid__add) ).

thf(4920,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1205]) ).

thf(51,axiom,
    class_Rings_Oring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1) ).

thf(1355,plain,
    class_Rings_Oring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(139,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring__no__zero__divisors @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
      <=> ( ( B
            = ( c_Groups_Ozero__class_Ozero @ C ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__0__iff) ).

thf(1580,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring__no__zero__divisors @ C )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            | ( A
              = ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
        & ( ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            | ( A
              = ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(1241,axiom,
    class_Rings_Olinordered__semiring__1__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__1__strict) ).

thf(4989,plain,
    class_Rings_Olinordered__semiring__1__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1241]) ).

thf(299,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Oadd__left) ).

thf(2133,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(769,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( ( hAPP @ K @ ( hAPP @ J @ I ) )
        = ( hAPP @ H @ ( hAPP @ G @ F ) ) )
     => ( ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ K @ J ) @ I )
        = ( hAPP @ ( c_Fun_Ocomp @ B @ D @ A @ H @ G ) @ F ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comp__cong) ).

thf(3752,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( ( hAPP @ K @ ( hAPP @ J @ I ) )
        = ( hAPP @ H @ ( hAPP @ G @ F ) ) )
     => ( ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ K @ J ) @ I )
        = ( hAPP @ ( c_Fun_Ocomp @ B @ D @ A @ H @ G ) @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(629,axiom,
    ( c_Complex_Oii
   != ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__i__not__zero) ).

thf(3239,plain,
    ( c_Complex_Oii
   != ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(668,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__power) ).

thf(3371,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(811,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) ).

thf(3884,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(601,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Oorder) ).

thf(3134,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(993,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__leD1) ).

thf(4493,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(212,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__refl) ).

thf(1824,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(930,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__diff__right) ).

thf(4284,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(305,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__nonneg__pos) ).

thf(2150,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(59,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__1) ).

thf(1370,plain,
    class_Rings_Ocomm__semiring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(527,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Ozero__class_Ozero @ A ) @ ( c_Groups_Oone__class_Oone @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__one) ).

thf(2890,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Ozero__class_Ozero @ A ) @ ( c_Groups_Oone__class_Oone @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(1153,axiom,
    class_RealVector_Ometric__space @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Ometric__space) ).

thf(4836,plain,
    class_RealVector_Ometric__space @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).

thf(888,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) )
      <=> ? [C: $i] :
          ! [D: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_lemma__NBseq__def2) ).

thf(4160,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) )
         => ? [C: $i] :
            ! [D: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) ) ) )
        & ( ? [C: $i] :
            ! [D: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) ) )
         => ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(422,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
          = A )
      <=> ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__equation__iff) ).

thf(2528,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
            = A )
         => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A )
            = B ) )
        & ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A )
            = B )
         => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(745,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( c_SEQ_OCauchy @ C @ ( c_COMBB @ C @ C @ tc_Nat_Onat @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_OCauchy) ).

thf(3664,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( c_SEQ_OCauchy @ C @ ( c_COMBB @ C @ C @ tc_Nat_Onat @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(776,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__inject__exp) ).

thf(3778,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(267,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ A )
     => ( class_Rings_Ocomm__ring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring__1) ).

thf(2015,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ A )
     => ( class_Rings_Ocomm__ring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(643,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__right__mono) ).

thf(3286,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(134,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) ) @ B )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__diff__inverse2) ).

thf(1560,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ ( c_Groups_Ozero__class_Ozero @ D ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
        <=> ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__cancel__left__neg) ).

thf(1938,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ ( c_Groups_Ozero__class_Ozero @ D ) )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
           => ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B )
           => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(1161,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) ).

thf(4849,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1161]) ).

thf(1026,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ ( c_Complex_Orcis @ C @ B ) ) @ A )
      = ( c_Complex_Orcis @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_DeMoivre2) ).

thf(4608,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ ( c_Complex_Orcis @ C @ B ) ) @ A )
      = ( c_Complex_Orcis @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(394,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( ( c_Orderings_Oord__class_Oless @ D @ A @ ( c_Groups_Oone__class_Oone @ D ) )
           => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__decreasing) ).

thf(2433,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( ( c_Orderings_Oord__class_Oless @ D @ A @ ( c_Groups_Oone__class_Oone @ D ) )
           => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(587,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ono__zero__divisors @ C )
     => ( ( B
         != ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( A
           != ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
           != ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_no__zero__divisors) ).

thf(3090,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ono__zero__divisors @ C )
     => ( ( B
         != ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( A
           != ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
           != ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(969,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Groups_Ozero__class_Ozero @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__zero__sym) ).

thf(4417,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( ( c_Groups_Ozero__class_Ozero @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Ozero__class_Ozero @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(783,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Omonoid__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__mult) ).

thf(3799,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Omonoid__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(938,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__increasing) ).

thf(4316,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(865,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__0__eq__0) ).

thf(4063,plain,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(642,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_odd__nonzero) ).

thf(3282,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(93,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
      | ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__linear) ).

thf(1423,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
      | ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

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

thf(3948,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(638,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__imp__0__less) ).

thf(3271,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(609,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
        <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__increasing__iff) ).

thf(3166,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
           => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
           => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(215,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oord @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Oord) ).

thf(1837,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oord @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(145,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__pos__pos) ).

thf(1602,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(457,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__mult__pos) ).

thf(2663,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(363,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ~ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__sum__squares__lt__zero) ).

thf(2332,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ~ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(505,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__div__algebra @ C )
     => ( ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__power) ).

thf(2826,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__div__algebra @ C )
     => ( ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(1218,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OBseq @ C @ ( c_COMBB @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) ) )
       => ( c_SEQ_OBseq @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__offset) ).

thf(4947,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OBseq @ C @ ( c_COMBB @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) ) )
       => ( c_SEQ_OBseq @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1218]) ).

thf(581,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__triangle__ineq2__sym) ).

thf(3065,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(768,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonpos__nonneg) ).

thf(3750,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(521,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i,D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ A ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ A @ C ) ) @ ( c_RealVector_Onorm__class_Onorm @ A @ D ) ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ononneg__bounded) ).

thf(2872,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i,D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ A ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ A @ C ) ) @ ( c_RealVector_Onorm__class_Onorm @ A @ D ) ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(1258,axiom,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(5021,plain,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1258]) ).

thf(233,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( A
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_equal__neg__zero) ).

thf(1888,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( A
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(18,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__comm__monoid__add) ).

thf(1296,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(1005,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__le__D2) ).

thf(4553,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).

thf(282,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__diff__add__eq2) ).

thf(2068,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(210,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ E )
     => ( ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Oprod__diff__prod) ).

thf(1819,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ E )
     => ( ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(604,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ A ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ? [C: $i] :
            ( ( B
              = ( c_Nat_OSuc @ C ) )
            & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__Suc__eq__0__disj) ).

thf(3144,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ A ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ? [C: $i] :
              ( ( B
                = ( c_Nat_OSuc @ C ) )
              & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ? [C: $i] :
              ( ( B
                = ( c_Nat_OSuc @ C ) )
              & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(1101,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring__0) ).

thf(4746,plain,
    class_Rings_Ocomm__semiring__0 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).

thf(964,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) ).

thf(4402,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(199,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E @ A )
         => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E @ C )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_unique__quotient__lemma__neg) ).

thf(1787,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E @ A )
         => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E @ C )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(1215,axiom,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(4942,plain,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1215]) ).

thf(1237,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ B ) ) @ A )
      = ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__of__real__power) ).

thf(4979,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ B ) ) @ A )
      = ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1237]) ).

thf(163,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__mono) ).

thf(1656,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(327,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) ).

thf(2211,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(82,axiom,
    class_Groups_Oordered__comm__monoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__comm__monoid__add) ).

thf(1407,plain,
    class_Groups_Oordered__comm__monoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(368,axiom,
    ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__one) ).

thf(2344,plain,
    ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(411,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Omult__zero @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Omult__zero) ).

thf(2492,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Omult__zero @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(762,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ominus @ E )
     => ( ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ E ) @ C @ B ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__apply) ).

thf(3727,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ominus @ E )
     => ( ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ E ) @ C @ B ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(13,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__left__commute) ).

thf(1281,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(983,axiom,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Ozero) ).

thf(4461,plain,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(1094,axiom,
    ? [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
      & ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Nat_Onat @ tc_RealDef_Oreal @ tc_Nat_Onat @ ( c_Fun_Ocomp @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ v_s ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact__096EX_Af_O_Asubseq_Af_A_G_Amonoseq_A_I_Fn_O_A_IRe_Ao_As_J_A_If_An_J_J_096) ).

thf(4732,plain,
    ? [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
      & ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Nat_Onat @ tc_RealDef_Oreal @ tc_Nat_Onat @ ( c_Fun_Ocomp @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ v_s ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).

thf(333,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__add__minus__iff) ).

thf(2228,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(740,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__assoc) ).

thf(3647,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(16,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__Suc) ).

thf(1291,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(570,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__imp__le__left) ).

thf(3033,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(185,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A )
            & ! [D: $i] :
                ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
               => ~ ( hBOOL @ ( hAPP @ B @ D ) ) )
            & ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ex__least__nat__less) ).

thf(1740,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A )
            & ! [D: $i] :
                ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
               => ~ ( hBOOL @ ( hAPP @ B @ D ) ) )
            & ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(79,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower_Opower @ E @ D @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower_Opower @ E @ D @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power_Opower_Opower__Suc) ).

thf(1402,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower_Opower @ E @ D @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower_Opower @ E @ D @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(5,axiom,
    class_Rings_Ocomm__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring) ).

thf(1270,plain,
    class_Rings_Ocomm__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(1232,axiom,
    class_Rings_Oring__no__zero__divisors @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors) ).

thf(4974,plain,
    class_Rings_Oring__no__zero__divisors @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1232]) ).

thf(916,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_n__less__m__mult__n) ).

thf(4248,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

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

thf(1375,plain,
    class_Rings_Oidom @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(94,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__less__iff__less) ).

thf(1425,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) )
        & ( ( c_Orderings_Oord__class_Oless @ C @ A @ B )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(417,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( C = A )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I2_J) ).

thf(2509,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( C = A )
         => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(873,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oabs__if @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
        & ( ~ ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
            = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__if) ).

thf(4095,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oabs__if @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
        & ( ~ ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(451,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( ( c_Groups_Oabs__class_Oabs @ A @ ( c_Groups_Oone__class_Oone @ A ) )
        = ( c_Groups_Oone__class_Oone @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__one) ).

thf(2638,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( ( c_Groups_Oabs__class_Oabs @ A @ ( c_Groups_Oone__class_Oone @ A ) )
        = ( c_Groups_Oone__class_Oone @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(337,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I5_J) ).

thf(2245,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(48,axiom,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) ).

thf(1350,plain,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(836,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Oincseq @ B @ A )
      <=> ! [C: $i,D: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
           => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) @ ( hAPP @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_incseq__def) ).

thf(3968,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( ( c_SEQ_Oincseq @ B @ A )
         => ! [C: $i,D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
             => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) @ ( hAPP @ A @ D ) ) ) )
        & ( ! [C: $i,D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
             => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) @ ( hAPP @ A @ D ) ) )
         => ( c_SEQ_Oincseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(1138,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__le__refl) ).

thf(4808,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1138]) ).

thf(755,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__right__mono) ).

thf(3699,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(354,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( ( c_Transcendental_Oln @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
      <=> ( A
          = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__eq__zero__iff) ).

thf(2300,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( ( ( c_Transcendental_Oln @ A )
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
         => ( A
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
        & ( ( A
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
         => ( ( c_Transcendental_Oln @ A )
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(777,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__Suc) ).

thf(3784,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(795,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__asym_H) ).

thf(3831,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(308,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__not__less__zero) ).

thf(2157,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(699,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__0) ).

thf(3482,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(190,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ouminus @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ ( tc_fun @ C @ D ) ) @ B ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ D ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_fun__Compl__def) ).

thf(1762,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ouminus @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ ( tc_fun @ C @ D ) ) @ B ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ D ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(127,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) )
      <=> ( ( B
           != ( c_Groups_Ozero__class_Ozero @ C ) )
          | ( A
           != ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sum__squares__gt__zero__iff) ).

thf(1534,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) )
         => ( ( B
             != ( c_Groups_Ozero__class_Ozero @ C ) )
            | ( A
             != ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
        & ( ( ( B
             != ( c_Groups_Ozero__class_Ozero @ C ) )
            | ( A
             != ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(202,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__1__neg) ).

thf(1794,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) )
         => ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(466,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_RComplete_Onatceiling @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatceiling @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__add__one) ).

thf(2685,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_RComplete_Onatceiling @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatceiling @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(432,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Rings_Ozero__neq__one @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ozero__neq__one) ).

thf(2577,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Rings_Ozero__neq__one @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(672,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__real__def) ).

thf(3381,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(862,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ C ) @ B )
              = A )
            & ! [D: $i] :
                ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ D )
                  & ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ D ) @ B )
                    = A ) )
               => ( D = C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__pos__nth__unique) ).

thf(4048,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ C ) @ B )
              = A )
            & ! [D: $i] :
                ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ D )
                  & ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ D ) @ B )
                    = A ) )
               => ( D = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(1003,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1 @ F )
     => ( ( c_Orderings_Oord__class_Oless__eq @ F @ E @ D )
       => ( ( c_Orderings_Oord__class_Oless__eq @ F @ C @ D )
         => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ B )
           => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ A )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convex__bound__le) ).

thf(4544,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1 @ F )
     => ( ( c_Orderings_Oord__class_Oless__eq @ F @ E @ D )
       => ( ( c_Orderings_Oord__class_Oless__eq @ F @ C @ D )
         => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ B )
           => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ A )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(972,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ C @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Complex__mult__complex__of__real) ).

thf(4429,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ C @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(1043,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re__complex__of__real) ).

thf(4636,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).

thf(117,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__0__le__iff__le) ).

thf(1498,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(179,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
      <=> ! [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
           => ? [D: $i] :
              ! [E: $i] :
                ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ E )
               => ! [F: $i] :
                    ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ F )
                   => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( c_Groups_Ominus__class_Ominus @ B @ ( hAPP @ A @ E ) @ ( hAPP @ A @ F ) ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Cauchy__iff) ).

thf(1719,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OCauchy @ B @ A )
         => ! [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
             => ? [D: $i] :
                ! [E: $i] :
                  ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ E )
                 => ! [F: $i] :
                      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ F )
                     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( c_Groups_Ominus__class_Ominus @ B @ ( hAPP @ A @ E ) @ ( hAPP @ A @ F ) ) ) @ C ) ) ) ) )
        & ( ! [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
             => ? [D: $i] :
                ! [E: $i] :
                  ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ E )
                 => ! [F: $i] :
                      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ F )
                     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( c_Groups_Ominus__class_Ominus @ B @ ( hAPP @ A @ E ) @ ( hAPP @ A @ F ) ) ) @ C ) ) ) )
         => ( c_SEQ_OCauchy @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(907,axiom,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Opos__bounded) ).

thf(4227,plain,
    ? [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      & ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(424,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__div__algebra @ C )
     => ( ( c_Groups_Osgn__class_Osgn @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Osgn__class_Osgn @ C @ B ) ) @ ( c_Groups_Osgn__class_Osgn @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__mult) ).

thf(2537,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__div__algebra @ C )
     => ( ( c_Groups_Osgn__class_Osgn @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Osgn__class_Osgn @ C @ B ) ) @ ( c_Groups_Osgn__class_Osgn @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(741,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__diff) ).

thf(3650,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(850,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__0__eq) ).

thf(4012,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(628,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( c_SEQ_OCauchy @ C @ ( c_COMBC @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_OCauchy) ).

thf(3237,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( c_SEQ_OCauchy @ C @ ( c_COMBC @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(830,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
      <=> ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
          & ( B != A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__le) ).

thf(3941,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ( B != A ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ( B != A ) )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(344,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_self__quotient__aux2) ).

thf(2268,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(617,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).

thf(3205,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(905,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__less__real__le) ).

thf(4219,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(566,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__mult__distrib2) ).

thf(3018,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(1191,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__mult) ).

thf(4898,plain,
    class_Groups_Oab__semigroup__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).

thf(1062,axiom,
    c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ v_f____ ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_f_I2_J) ).

thf(4676,plain,
    c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ v_f____ ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(1053,axiom,
    class_Rings_Oordered__comm__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__comm__semiring) ).

thf(4653,plain,
    class_Rings_Oordered__comm__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).

thf(270,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__add__less__zero__iff__single__add__less__zero) ).

thf(2023,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(1264,axiom,
    ! [A: $i] :
      ( ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Oconvergent) ).

thf(5031,plain,
    ! [A: $i] :
      ( ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1264]) ).

thf(55,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ominus @ B )
     => ( class_Groups_Ominus @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Groups_Ominus) ).

thf(1361,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ominus @ B )
     => ( class_Groups_Ominus @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(1017,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__nonneg__nonneg) ).

thf(4582,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(727,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__1) ).

thf(3596,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(896,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__assoc2) ).

thf(4191,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(427,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__add__distrib) ).

thf(2551,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(1171,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) @ ( c_RComplete_Onatceiling @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__mono) ).

thf(4867,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) @ ( c_RComplete_Onatceiling @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1171]) ).

thf(982,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Odecseq @ B @ A )
      <=> ! [C: $i,D: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
           => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) @ ( hAPP @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_decseq__def) ).

thf(4457,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( ( c_SEQ_Odecseq @ B @ A )
         => ! [C: $i,D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
             => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) @ ( hAPP @ A @ C ) ) ) )
        & ( ! [C: $i,D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
             => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) @ ( hAPP @ A @ C ) ) )
         => ( c_SEQ_Odecseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(403,axiom,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ( c_Lim_OisUCont @ tc_RealDef_Oreal @ A @ ( c_RealVector_Oof__real @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_OisUCont) ).

thf(2460,plain,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ( c_Lim_OisUCont @ tc_RealDef_Oreal @ A @ ( c_RealVector_Oof__real @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(115,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Oadd) ).

thf(1493,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(940,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__ge__zero) ).

thf(4320,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(1221,axiom,
    class_Rings_Osemiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Osemiring) ).

thf(4951,plain,
    class_Rings_Osemiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1221]) ).

thf(652,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Olinordered__ab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add) ).

thf(3311,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Olinordered__ab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(257,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Olinorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Olinorder) ).

thf(1964,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Olinorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(823,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__add__mult__distrib) ).

thf(3922,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(679,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__triangle__ineq3) ).

thf(3407,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(431,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    <=> ( ( ( B
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        | ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__is__1) ).

thf(2561,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( ( B
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) )
      & ( ( ( ( B
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(1120,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__ab__semigroup__add) ).

thf(4772,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).

thf(1088,axiom,
    ( ( hAPP @ c_Complex_ORe @ c_Complex_Oii )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__Re__i) ).

thf(4721,plain,
    ( ( hAPP @ c_Complex_ORe @ c_Complex_Oii )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).

thf(105,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( C = A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I4_J) ).

thf(1461,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( C = A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(690,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__less__add1) ).

thf(3441,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(357,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__sgn) ).

thf(2312,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(401,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__mult__distrib) ).

thf(2455,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(1104,axiom,
    ! [A: $i,B: $i] :
      ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Complex_Orcis @ B @ A ) )
      = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__mod__rcis) ).

thf(4750,plain,
    ! [A: $i,B: $i] :
      ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Complex_Orcis @ B @ A ) )
      = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).

thf(86,axiom,
    class_Rings_Oordered__cancel__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__cancel__semiring) ).

thf(1414,plain,
    class_Rings_Oordered__cancel__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(322,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__le__eq__diff) ).

thf(2196,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(335,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__1__mult) ).

thf(2240,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(1099,axiom,
    class_Rings_Oordered__cancel__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__cancel__semiring) ).

thf(4744,plain,
    class_Rings_Oordered__cancel__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).

thf(444,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( A != B )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I11_J) ).

thf(2613,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( A != B )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(1050,axiom,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__not__refl) ).

thf(4648,plain,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(207,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__bin__lemma) ).

thf(1810,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(201,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__gt__zero__iff) ).

thf(1791,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(149,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Omonoseq @ B @ A )
      <=> ( ! [C: $i,D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
             => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) @ ( hAPP @ A @ D ) ) )
          | ! [C: $i,D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
             => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) @ ( hAPP @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monoseq__def) ).

thf(1612,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( ( c_SEQ_Omonoseq @ B @ A )
         => ( ! [C: $i,D: $i] :
                ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
               => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) @ ( hAPP @ A @ D ) ) )
            | ! [C: $i,D: $i] :
                ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
               => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) @ ( hAPP @ A @ C ) ) ) ) )
        & ( ( ! [C: $i,D: $i] :
                ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
               => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) @ ( hAPP @ A @ D ) ) )
            | ! [C: $i,D: $i] :
                ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ D )
               => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) @ ( hAPP @ A @ C ) ) ) )
         => ( c_SEQ_Omonoseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(406,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ominus__right) ).

thf(2472,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(1033,axiom,
    class_Rings_Ocomm__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring) ).

thf(4621,plain,
    class_Rings_Ocomm__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(455,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( class_Power_Opower @ C )
        & ( class_Rings_Omult__zero @ C )
        & ( class_Rings_Ono__zero__divisors @ C )
        & ( class_Rings_Ozero__neq__one @ C ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
      <=> ( ( B
            = ( c_Groups_Ozero__class_Ozero @ C ) )
          & ( A
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__eq__0__iff) ).

thf(2649,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( class_Power_Opower @ C )
        & ( class_Rings_Omult__zero @ C )
        & ( class_Rings_Ono__zero__divisors @ C )
        & ( class_Rings_Ozero__neq__one @ C ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( A
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        & ( ( ( B
              = ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( A
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(893,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
      <=> ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__def) ).

thf(4180,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OBseq @ B @ A )
         => ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) ) )
        & ( ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) )
         => ( c_SEQ_OBseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(1096,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__le__iff) ).

thf(4736,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).

thf(186,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__ge__zero__imp__ge__one) ).

thf(1744,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(494,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Complex_Ocomplex_OComplex @ B @ A )
        = c_Complex_Oii )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
        & ( A
          = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Complex__eq__i) ).

thf(2781,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Complex_Ocomplex_OComplex @ B @ A )
          = c_Complex_Oii )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( ( c_Complex_Ocomplex_OComplex @ B @ A )
          = c_Complex_Oii ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(1000,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__of__real__def) ).

thf(4530,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(91,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__assoc) ).

thf(1419,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(542,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__eq__0) ).

thf(2932,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( ( ( c_Groups_Oabs__class_Oabs @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(512,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__mono1) ).

thf(2845,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(247,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) ).

thf(1932,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(413,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) )
      <=> ( A
         != ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__norm__iff) ).

thf(2497,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) )
         => ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(227,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ E @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) ) @ ( c_RealVector_Onorm__class_Onorm @ E @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__diff__triangle__ineq) ).

thf(1874,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ E @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) ) @ ( c_RealVector_Onorm__class_Onorm @ E @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(921,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).

thf(4259,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(484,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D @ F )
         => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
           => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
             => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ F )
               => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__mono2__neg__lemma) ).

thf(2756,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D @ F )
         => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
           => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
             => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ F )
               => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(75,axiom,
    class_Rings_Oring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring) ).

thf(1393,plain,
    class_Rings_Oring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(1134,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__add) ).

thf(4804,plain,
    class_Groups_Ocomm__monoid__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).

thf(630,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__le__zero__iff) ).

thf(3242,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(757,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
        <=> ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__cancel__left__pos) ).

thf(3703,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
           => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
           => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(34,axiom,
    class_Groups_Ozero @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ozero) ).

thf(1324,plain,
    class_Groups_Ozero @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(348,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ D ) @ B @ A )
      <=> ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B @ A )
          & ~ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__fun__def) ).

thf(2285,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ D ) @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B @ A )
            & ~ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ A @ B ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B @ A )
            & ~ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ A @ B ) )
         => ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ D ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(420,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_even__less__0__iff) ).

thf(2519,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(958,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ocomm__neutral) ).

thf(4384,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(19,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__add) ).

thf(1297,plain,
    class_Groups_Ocomm__monoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(599,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
      <=> ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ? [D: $i] :
              ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ D ) ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__iff2) ).

thf(3126,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OBseq @ B @ A )
         => ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ? [D: $i] :
                ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ D ) ) ) @ C ) ) )
        & ( ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ? [D: $i] :
                ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ D ) ) ) @ C ) )
         => ( c_SEQ_OBseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(509,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__mult__self) ).

thf(2836,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(997,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
    <=> ~ ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
            & ~ ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          | ? [D: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
              & ~ ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__diff__split__asm) ).

thf(4508,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
       => ~ ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
              & ~ ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
            | ? [D: $i] :
                ( ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
                & ~ ( hBOOL @ ( hAPP @ C @ D ) ) ) ) )
      & ( ~ ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
              & ~ ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
            | ? [D: $i] :
                ( ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
                & ~ ( hBOOL @ ( hAPP @ C @ D ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(67,axiom,
    class_Groups_Osgn__if @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Osgn__if) ).

thf(1382,plain,
    class_Groups_Osgn__if @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(434,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) ).

thf(2581,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(405,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
    <=> ? [C: $i] :
          ( A
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__Suc__ex__iff) ).

thf(2465,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ? [C: $i] :
            ( A
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) )
      & ( ? [C: $i] :
            ( A
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(415,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convergent__realpow) ).

thf(2505,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(911,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__Im__le__cmod) ).

thf(4237,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(203,axiom,
    ! [A: $i] :
      ( ( c_RComplete_Onatceiling @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__real__of__nat) ).

thf(1799,plain,
    ! [A: $i] :
      ( ( c_RComplete_Onatceiling @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(1085,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add) ).

thf(4717,plain,
    class_Groups_Ocancel__semigroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).

thf(253,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos) ).

thf(1947,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(879,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( hAPP @ B @ D ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( hAPP @ B @ C ) )
         => ? [D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat0__intermed__int__val) ).

thf(4120,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( hAPP @ B @ D ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( hAPP @ B @ C ) )
         => ? [D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(220,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
    <=> ? [C: $i] :
          ( A
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__iff__add) ).

thf(1852,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ? [C: $i] :
            ( A
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) )
      & ( ? [C: $i] :
            ( A
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(501,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0__right) ).

thf(2813,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(772,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__natfloor__le) ).

thf(3760,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(680,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Omult__zero @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__zero__right) ).

thf(3409,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Omult__zero @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(883,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__add__cancel) ).

thf(4135,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(488,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ozero @ A )
     => ( class_Groups_Ozero @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ozero) ).

thf(2766,plain,
    ! [A: $i] :
      ( ( class_Groups_Ozero @ A )
     => ( class_Groups_Ozero @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(144,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_self__quotient__aux1) ).

thf(1599,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(1195,axiom,
    class_Rings_Oidom @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oidom) ).

thf(4902,plain,
    class_Rings_Oidom @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).

thf(261,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
    <=> ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
         => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        & ! [D: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__diff__split) ).

thf(1979,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          & ! [D: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
             => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) )
      & ( ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          & ! [D: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
             => ( hBOOL @ ( hAPP @ C @ D ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(1072,axiom,
    ! [A: $i] :
      ( ( ( c_RealDef_Oreal @ tc_Nat_Onat @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__zero__iff) ).

thf(4689,plain,
    ! [A: $i] :
      ( ( ( ( c_RealDef_Oreal @ tc_Nat_Onat @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( c_RealDef_Oreal @ tc_Nat_Onat @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(1261,axiom,
    class_RealVector_Oreal__normed__div__algebra @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra) ).

thf(5026,plain,
    class_RealVector_Oreal__normed__div__algebra @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1261]) ).

thf(867,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( c_Groups_Oone__class_Oone @ B ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__1__pos) ).

thf(4069,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Oone__class_Oone @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Oone__class_Oone @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(822,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Ocomm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__add) ).

thf(3920,plain,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Ocomm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(665,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__mono) ).

thf(3364,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(154,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ A ) @ B )
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ A ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__eq__if) ).

thf(1627,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ A ) @ B )
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ A ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(126,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oidom @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
      <=> ( ( B = A )
          | ( B
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__two__disj) ).

thf(1525,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oidom @ C )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
            = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
         => ( ( B = A )
            | ( B
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) )
        & ( ( ( B = A )
            | ( B
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
            = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(697,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) ).

thf(3476,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(770,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__iff1) ).

thf(3755,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(28,axiom,
    class_Groups_Ogroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ogroup__add) ).

thf(1318,plain,
    class_Groups_Ogroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(948,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__asym) ).

thf(4352,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(1047,axiom,
    class_Groups_Ozero @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ozero) ).

thf(4644,plain,
    class_Groups_Ozero @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).

thf(1037,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__le__eq__one) ).

thf(4625,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(1011,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) ).

thf(4567,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).

thf(803,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zpower__zadd__distrib) ).

thf(3856,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(135,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Power_Opower @ C )
     => ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ B )
            = ( c_Groups_Oone__class_Oone @ C ) ) )
        & ( ( B
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__num__eq__if) ).

thf(1563,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Power_Opower @ C )
     => ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ B )
            = ( c_Groups_Oone__class_Oone @ C ) ) )
        & ( ( B
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(846,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__power__less__imp__less) ).

thf(3999,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(384,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ B )
     => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__of__real) ).

thf(2402,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ B )
     => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(166,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__add__left__mono) ).

thf(1663,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(122,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Omonoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__add) ).

thf(1515,plain,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Omonoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(706,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__of__pos) ).

thf(3508,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(320,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
        | ( B = A ) )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__or__eq__imp__le) ).

thf(2189,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
        | ( B = A ) )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(799,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__triangle__ineq4) ).

thf(3841,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(848,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__cancel__less) ).

thf(4004,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(1245,axiom,
    class_Groups_Oordered__ab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) ).

thf(4995,plain,
    class_Groups_Oordered__ab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1245]) ).

thf(719,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Power_Opower @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( c_Groups_Oone__class_Oone @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__0) ).

thf(3563,plain,
    ! [A: $i,B: $i] :
      ( ( class_Power_Opower @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( c_Groups_Oone__class_Oone @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(486,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__idom @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__idom) ).

thf(2761,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__idom @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(641,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__mult__def) ).

thf(3279,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(1183,axiom,
    class_Orderings_Oord @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Oord) ).

thf(4889,plain,
    class_Orderings_Oord @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).

thf(1226,axiom,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_ORe,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_OisUCont) ).

thf(4966,plain,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_ORe,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1226]) ).

thf(175,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__gt__zero) ).

thf(1695,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(1113,axiom,
    class_Orderings_Olinorder @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Olinorder) ).

thf(4765,plain,
    class_Orderings_Olinorder @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).

thf(525,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__imp__neq) ).

thf(2883,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(989,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Oring__1__no__zero__divisors @ B )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
          = ( c_Groups_Oone__class_Oone @ B ) )
      <=> ( ( A
            = ( c_Groups_Oone__class_Oone @ B ) )
          | ( A
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_square__eq__1__iff) ).

thf(4476,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Oring__1__no__zero__divisors @ B )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
            = ( c_Groups_Oone__class_Oone @ B ) )
         => ( ( A
              = ( c_Groups_Oone__class_Oone @ B ) )
            | ( A
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) )
        & ( ( ( A
              = ( c_Groups_Oone__class_Oone @ B ) )
            | ( A
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
            = ( c_Groups_Oone__class_Oone @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(537,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring) ).

thf(2915,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(979,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Odiff__right) ).

thf(4448,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(160,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Omonoid__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__add) ).

thf(1648,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Omonoid__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(387,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__le__add__iff1) ).

thf(2413,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(949,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__add__less__0__iff) ).

thf(4354,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(1022,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__strict__mono_H) ).

thf(4594,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).

thf(1176,axiom,
    class_Groups_Omonoid__mult @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Omonoid__mult) ).

thf(4874,plain,
    class_Groups_Omonoid__mult @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).

thf(702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__le__add2) ).

thf(3491,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(130,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
          | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) ) )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_split__mult__neg__le) ).

thf(1548,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
          | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) ) )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(1056,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
       => ( c_SEQ_OBseq @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Cauchy__Bseq) ).

thf(4657,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
       => ( c_SEQ_OBseq @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).

thf(961,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Orcis @ D @ C ) ) @ ( c_Complex_Orcis @ B @ A ) )
      = ( c_Complex_Orcis @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_rcis__mult) ).

thf(4393,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Orcis @ D @ C ) ) @ ( c_Complex_Orcis @ B @ A ) )
      = ( c_Complex_Orcis @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(1129,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
      <=> ( c_SEQ_OBseq @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__minus__iff) ).

thf(4797,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OBseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( c_SEQ_OBseq @ B @ A ) )
        & ( ( c_SEQ_OBseq @ B @ A )
         => ( c_SEQ_OBseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).

thf(950,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__0__right) ).

thf(4359,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(751,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib2) ).

thf(3686,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(992,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I3_J) ).

thf(4489,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(334,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mult__less__cancel1) ).

thf(2235,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(302,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Groups_Oab__semigroup__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__mult) ).

thf(2144,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Groups_Oab__semigroup__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(1206,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) @ ( c_RComplete_Onatfloor @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__mono) ).

thf(4921,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) @ ( c_RComplete_Onatfloor @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1206]) ).

thf(10,axiom,
    class_Rings_Olinordered__semiring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1) ).

thf(1278,plain,
    class_Rings_Olinordered__semiring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(316,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) @ A )
    <=> ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) @ B )
        & ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__le__interval__iff) ).

thf(2176,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) @ B )
          & ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) @ B )
          & ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(1081,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__le__trans) ).

thf(4708,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).

thf(1069,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4685,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(350,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__less__mono2) ).

thf(2292,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(1174,axiom,
    class_RealVector_Ometric__space @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Ometric__space) ).

thf(4872,plain,
    class_RealVector_Ometric__space @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).

thf(533,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = ( c_Groups_Oone__class_Oone @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__one) ).

thf(2904,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = ( c_Groups_Oone__class_Oone @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(807,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_incr__mult__lemma) ).

thf(3871,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(273,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
          = A )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__equal__zero) ).

thf(2033,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
            = A )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(946,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Ominus) ).

thf(4340,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(798,axiom,
    ( c_Complex_Oii
    = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_i__def) ).

thf(3839,plain,
    ( c_Complex_Oii
    = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(712,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).

thf(3523,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(477,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zminus__inverse2) ).

thf(2734,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(1254,axiom,
    c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ v_f____ @ v_ga____ ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_g_I2_J) ).

thf(5016,plain,
    c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ v_f____ @ v_ga____ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1254]) ).

thf(1015,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) @ ( c_Nat_OSuc @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__less__Suc) ).

thf(4577,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) @ ( c_Nat_OSuc @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).

thf(529,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__square__less__zero) ).

thf(2894,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(952,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__group__add__abs @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__group__add__abs) ).

thf(4365,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__group__add__abs @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(1216,axiom,
    class_Groups_Ominus @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ominus) ).

thf(4943,plain,
    class_Groups_Ominus @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1216]) ).

thf(859,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__gt1__lemma) ).

thf(4041,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(1058,axiom,
    ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__1) ).

thf(4662,plain,
    ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(583,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__is__0) ).

thf(3070,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
      <=> ( C = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__right__cancel) ).

thf(1916,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
         => ( C = A ) )
        & ( ( C = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(231,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__funE) ).

thf(1884,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(788,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__trans) ).

thf(3811,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(710,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__neg__neg) ).

thf(3519,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(410,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__idempotent) ).

thf(2489,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(1025,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__idem) ).

thf(4605,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).

thf(943,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
      <=> ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__not__less) ).

thf(4328,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(957,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Complex_Oexpi @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Oexpi @ B ) ) @ ( c_Complex_Oexpi @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_expi__add) ).

thf(4381,plain,
    ! [A: $i,B: $i] :
      ( ( c_Complex_Oexpi @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Oexpi @ B ) ) @ ( c_Complex_Oexpi @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(728,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__imp__not__eq) ).

thf(3599,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(553,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
         => ? [D: $i] :
            ! [E: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ E )
             => ! [F: $i] :
                  ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ F )
                 => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ B @ E ) @ ( hAPP @ B @ F ) ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_CauchyD) ).

thf(2970,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
         => ? [D: $i] :
            ! [E: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ E )
             => ! [F: $i] :
                  ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ F )
                 => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ B @ E ) @ ( hAPP @ B @ F ) ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(654,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ C @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Complex__add__complex__of__real) ).

thf(3319,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ C @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(627,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1__strict @ F )
     => ( ( c_Orderings_Oord__class_Oless @ F @ E @ D )
       => ( ( c_Orderings_Oord__class_Oless @ F @ C @ D )
         => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ B )
           => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ A )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convex__bound__lt) ).

thf(3234,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1__strict @ F )
     => ( ( c_Orderings_Oord__class_Oless @ F @ E @ D )
       => ( ( c_Orderings_Oord__class_Oless @ F @ C @ D )
         => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ B )
           => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) @ A )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(277,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Odiff__left) ).

thf(2048,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(824,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__triangle__ineq3) ).

thf(3925,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(517,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oring @ F )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
      <=> ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__add__iff1) ).

thf(2859,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oring @ F )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C )
            = A ) )
        & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C )
            = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(543,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
      <=> ( ( c_Orderings_Oord__class_Oless @ C @ A @ B )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__less__iff__gr__or__eq) ).

thf(2938,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless @ C @ A @ B )
            | ( B = A ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless @ C @ A @ B )
            | ( B = A ) )
         => ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(198,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( c_Groups_Oone__class_Oone @ C ) ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__minus) ).

thf(1784,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( c_Groups_Oone__class_Oone @ C ) ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(73,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( class_Orderings_Oorder @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Orderings_Oorder) ).

thf(1389,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( class_Orderings_Oorder @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(556,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__commute) ).

thf(2981,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(935,axiom,
    c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__0__less__1) ).

thf(4309,plain,
    c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(771,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__triangle__ineq2) ).

thf(3758,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(246,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Complex_Ocomplex_Ocomplex__case @ D @ C @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex_Osimps_I2_J) ).

thf(1929,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Complex_Ocomplex_Ocomplex__case @ D @ C @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(1208,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ D @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__add) ).

thf(4926,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ D @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1208]) ).

thf(1063,axiom,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(4677,plain,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(636,axiom,
    ? [A: $i] :
    ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Obounded) ).

thf(3266,plain,
    ? [A: $i] :
    ! [B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(462,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__natfloor) ).

thf(2674,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(72,axiom,
    class_Power_Opower @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Power_Opower) ).

thf(1388,plain,
    class_Power_Opower @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(1131,axiom,
    class_Groups_Oordered__ab__group__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__group__add) ).

thf(4801,plain,
    class_Groups_Oordered__ab__group__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).

thf(1020,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__ge__zero) ).

thf(4590,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).

thf(1233,axiom,
    class_Groups_Ozero @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ozero) ).

thf(4975,plain,
    class_Groups_Ozero @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1233]) ).

thf(678,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( ( c_RComplete_Onatceiling @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__eq) ).

thf(3404,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( ( c_RComplete_Onatceiling @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(1028,axiom,
    ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Ozero) ).

thf(4612,plain,
    ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).

thf(1117,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add) ).

thf(4769,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).

thf(1259,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Ominus) ).

thf(5022,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1259]) ).

thf(113,axiom,
    ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zminus__0) ).

thf(1488,plain,
    ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(550,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( B
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__neg__iff__add__eq__0) ).

thf(2959,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( B
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) )
        & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( B
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(620,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
        & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__0__less__mult__iff) ).

thf(3213,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          & ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(782,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gr0I) ).

thf(3796,plain,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(990,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_RComplete_Onatceiling @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__natceiling) ).

thf(4485,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_RComplete_Onatceiling @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(1007,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Oone__class_Oone @ C ) )
           => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__le__one__le) ).

thf(4558,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Oone__class_Oone @ C ) )
           => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(1236,axiom,
    class_Rings_Oidom @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oidom) ).

thf(4978,plain,
    class_Rings_Oidom @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1236]) ).

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

thf(2326,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(578,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__ge__zero) ).

thf(3051,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(658,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__cancel__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring) ).

thf(3330,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__cancel__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(618,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add_Ocomm__neutral) ).

thf(3208,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(967,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oring__no__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors) ).

thf(4413,plain,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oring__no__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(37,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
      | ( B = A )
      | ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zless__linear) ).

thf(1332,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
      | ( B = A )
      | ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(864,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).

thf(4060,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(650,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = B )
     => ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__eq__self__zero) ).

thf(3304,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = B )
     => ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(95,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Omult__zero @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__zero__left) ).

thf(1428,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Omult__zero @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(623,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__left) ).

thf(3223,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(596,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__le__add__iff2) ).

thf(3118,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(1265,axiom,
    ! [A: $i,B: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ B )
        & ( class_RealVector_Oreal__normed__vector @ B ) )
     => ( ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A )
       => ( c_SEQ_Oconvergent @ B @ ( c_COMBB @ tc_RealDef_Oreal @ B @ tc_Nat_Onat @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Oconvergent) ).

thf(5033,plain,
    ! [A: $i,B: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ B )
        & ( class_RealVector_Oreal__normed__vector @ B ) )
     => ( ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A )
       => ( c_SEQ_Oconvergent @ B @ ( c_COMBB @ tc_RealDef_Oreal @ B @ tc_Nat_Onat @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1265]) ).

thf(880,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Ometric__space @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( ( c_SEQ_Osubseq @ A )
         => ( c_SEQ_OCauchy @ C @ ( c_Fun_Ocomp @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Cauchy__subseq__Cauchy) ).

thf(4127,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Ometric__space @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( ( c_SEQ_Osubseq @ A )
         => ( c_SEQ_OCauchy @ C @ ( c_Fun_Ocomp @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(329,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__imp__le__nat) ).

thf(2220,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(1128,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__mult) ).

thf(4796,plain,
    class_Groups_Oab__semigroup__mult @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).

thf(6,axiom,
    class_Groups_Oab__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__add) ).

thf(1271,plain,
    class_Groups_Oab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(60,axiom,
    class_Rings_Oring__no__zero__divisors @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__no__zero__divisors) ).

thf(1371,plain,
    class_Rings_Oring__no__zero__divisors @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(1154,axiom,
    class_Rings_Olinordered__semidom @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semidom) ).

thf(4837,plain,
    class_Rings_Olinordered__semidom @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).

thf(1048,axiom,
    class_Rings_Oordered__ring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__ring) ).

thf(4645,plain,
    class_Rings_Oordered__ring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).

thf(155,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Osemiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring) ).

thf(1635,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Osemiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(1146,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( c_SEQ_Odecseq @ B @ A )
      <=> ( c_SEQ_Oincseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_decseq__eq__incseq) ).

thf(4822,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( c_SEQ_Odecseq @ B @ A )
         => ( c_SEQ_Oincseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
        & ( ( c_SEQ_Oincseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( c_SEQ_Odecseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).

thf(182,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__mod__triangle__sub) ).

thf(1733,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(263,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ C )
           => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__less__cases) ).

thf(1997,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ C )
           => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(662,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add1__zle__eq) ).

thf(3340,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(726,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I1_J) ).

thf(3594,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(483,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( B != A )
       => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__neqE__linordered__idom) ).

thf(2753,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( B != A )
       => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(397,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ( c_Fun_Ocomp @ I @ H @ G @ F @ E )
        = ( c_Fun_Ocomp @ D @ H @ G @ C @ B ) )
     => ( ( hAPP @ F @ ( hAPP @ E @ A ) )
        = ( hAPP @ C @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_o__eq__dest) ).

thf(2442,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ( c_Fun_Ocomp @ I @ H @ G @ F @ E )
        = ( c_Fun_Ocomp @ D @ H @ G @ C @ B ) )
     => ( ( hAPP @ F @ ( hAPP @ E @ A ) )
        = ( hAPP @ C @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(485,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oidom) ).

thf(2759,plain,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(312,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__semigroup__add__imp__le @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(2166,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__semigroup__add__imp__le @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(191,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__mono__neg) ).

thf(1765,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(734,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__add__one__self__le__self) ).

thf(3619,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(235,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oring__1__no__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors) ).

thf(1897,plain,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oring__1__no__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(429,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add) ).

thf(2557,plain,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(1045,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Ominus) ).

thf(4640,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).

thf(1181,axiom,
    c_SEQ_Osubseq @ v_f____,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_f_I1_J) ).

thf(4887,plain,
    c_SEQ_Osubseq @ v_f____,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).

thf(915,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ A )
     => ( ( c_Groups_Oabs__class_Oabs @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__zero) ).

thf(4245,plain,
    ! [A: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ A )
     => ( ( c_Groups_Oabs__class_Oabs @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(304,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ A ) ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__less) ).

thf(2148,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(360,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( B
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( A
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_equation__minus__iff) ).

thf(2319,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( B
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( A
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) )
        & ( ( A
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) )
         => ( B
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(52,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(1356,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(369,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__natfloor) ).

thf(2346,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(295,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) ).

thf(2119,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(1209,axiom,
    ! [A: $i] :
      ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__real__of__nat__cancel) ).

thf(4929,plain,
    ! [A: $i] :
      ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1209]) ).

thf(790,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) ).

thf(3815,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(717,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
      <=> ( B
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__eq__add__iff2) ).

thf(3551,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
         => ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) )
        & ( ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(1032,axiom,
    class_Rings_Oordered__cancel__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__cancel__semiring) ).

thf(4620,plain,
    class_Rings_Oordered__cancel__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).

thf(684,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__left__mono) ).

thf(3419,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(128,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) @ A )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__le__eq) ).

thf(1542,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(499,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__eq__less__or__eq) ).

thf(2802,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
          | ( B = A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
          | ( B = A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(259,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__if) ).

thf(1969,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(591,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( B
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        & ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__1__eq__mult__iff) ).

thf(3101,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
       => ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(1126,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ A ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__natfloor__eq__one) ).

thf(4790,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).

thf(464,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__of__neg) ).

thf(2679,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(1078,axiom,
    class_Orderings_Opreorder @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Opreorder) ).

thf(4704,plain,
    class_Orderings_Opreorder @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).

thf(845,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__of__nonpos) ).

thf(3996,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(83,axiom,
    class_Groups_Oabs__if @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oabs__if) ).

thf(1408,plain,
    class_Groups_Oabs__if @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(995,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ozero__right) ).

thf(4498,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(408,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__eq__nat_Osimps_I1_J) ).

thf(2478,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(632,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Oincseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_incseq__imp__monoseq) ).

thf(3251,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Oincseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(954,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ C )
         => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ D )
           => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
             => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ F )
               => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ E @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiv__mono2__lemma) ).

thf(4369,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ C )
         => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ D )
           => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
             => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C @ F )
               => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ E @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(328,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__left__cancel) ).

thf(2214,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(380,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_i__complex__of__real) ).

thf(2383,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(448,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
        | ( B = A )
        | ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__less__linear) ).

thf(2624,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
        | ( B = A )
        | ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(234,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__0) ).

thf(1894,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(558,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neq0__conv) ).

thf(2987,plain,
    ! [A: $i] :
      ( ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(1201,axiom,
    class_Rings_Oring__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1) ).

thf(4914,plain,
    class_Rings_Oring__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).

thf(1075,axiom,
    class_Rings_Oring__no__zero__divisors @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__no__zero__divisors) ).

thf(4699,plain,
    class_Rings_Oring__no__zero__divisors @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(764,axiom,
    ! [A: $i] :
      ( ( c_RComplete_Onatfloor @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__real__of__nat) ).

thf(3732,plain,
    ! [A: $i] :
      ( ( c_RComplete_Onatfloor @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(342,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__not__sym) ).

thf(2257,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(140,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__increasing) ).

thf(1589,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(548,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__less__mono) ).

thf(2955,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(353,axiom,
    c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ v_r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_rp) ).

thf(2299,plain,
    c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ v_r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(1135,axiom,
    class_Groups_Omonoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__add) ).

thf(4805,plain,
    class_Groups_Omonoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).

thf(460,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__diff__conv2) ).

thf(2669,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) @ B ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(294,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A )
      = ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__real__of__nat) ).

thf(2116,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A )
      = ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(510,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__less__0__iff__less) ).

thf(2839,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(1224,axiom,
    class_Groups_Omonoid__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__add) ).

thf(4958,plain,
    class_Groups_Omonoid__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1224]) ).

thf(544,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__cancel2) ).

thf(2945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(903,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Ononneg__bounded) ).

thf(4213,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(1030,axiom,
    class_Rings_Oordered__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__semiring) ).

thf(4616,plain,
    class_Rings_Oordered__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(378,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ A )
      <=> ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__le__self__iff) ).

thf(2378,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(530,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Osemiring__0 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring__0) ).

thf(2896,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Osemiring__0 @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(673,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zabs__less__one__iff) ).

thf(3384,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(1121,axiom,
    class_Rings_Osemiring__0 @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring__0) ).

thf(4773,plain,
    class_Rings_Osemiring__0 @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).

thf(64,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__commute) ).

thf(1376,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(99,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_gr__implies__not0) ).

thf(1439,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(297,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__strict__right__mono) ).

thf(2129,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(1012,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__less__zero) ).

thf(4569,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).

thf(181,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__nonpos__neg) ).

thf(1731,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(398,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_o__def) ).

thf(2446,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(222,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__ring) ).

thf(1862,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(35,axiom,
    class_Rings_Oordered__ring__abs @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring__abs) ).

thf(1325,plain,
    class_Rings_Oordered__ring__abs @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(1162,axiom,
    class_Rings_Omult__zero @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Omult__zero) ).

thf(4850,plain,
    class_Rings_Omult__zero @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).

thf(45,axiom,
    class_Rings_Osemiring__0 @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring__0) ).

thf(1344,plain,
    class_Rings_Osemiring__0 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(349,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).

thf(2289,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(340,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_o__apply) ).

thf(2252,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(1242,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4990,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1242]) ).

thf(621,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__less__power__Suc) ).

thf(3219,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(504,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__semidom @ B )
     => ( c_Orderings_Oord__class_Oless @ B @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__one) ).

thf(2824,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__semidom @ B )
     => ( c_Orderings_Oord__class_Oless @ B @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(900,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ B ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_incr__lemma) ).

thf(4203,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ B ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(1042,axiom,
    class_Groups_Ominus @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ominus) ).

thf(4635,plain,
    class_Groups_Ominus @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(317,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__commutes) ).

thf(2182,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(1266,axiom,
    class_Rings_Osemiring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring) ).

thf(5035,plain,
    class_Rings_Osemiring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1266]) ).

thf(108,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__leD2) ).

thf(1471,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(849,axiom,
    ! [A: $i,B: $i] :
      ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__add) ).

thf(4009,plain,
    ! [A: $i,B: $i] :
      ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(875,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__diff__def) ).

thf(4103,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(133,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_right__minus) ).

thf(1557,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(1093,axiom,
    class_Orderings_Oorder @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oorder) ).

thf(4731,plain,
    class_Orderings_Oorder @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).

thf(855,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Ocomm__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring) ).

thf(4028,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Ocomm__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(239,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__less__add__iff2) ).

thf(1909,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(1190,axiom,
    class_Rings_Olinordered__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring) ).

thf(4897,plain,
    class_Rings_Olinordered__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).

thf(1239,axiom,
    class_RealVector_Oreal__normed__algebra @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__algebra) ).

thf(4985,plain,
    class_RealVector_Oreal__normed__algebra @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1239]) ).

thf(648,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Complex_ORe @ B )
        = ( hAPP @ c_Complex_ORe @ A ) )
     => ( ( ( hAPP @ c_Complex_OIm @ B )
          = ( hAPP @ c_Complex_OIm @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__equality) ).

thf(3298,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Complex_ORe @ B )
        = ( hAPP @ c_Complex_ORe @ A ) )
     => ( ( ( hAPP @ c_Complex_OIm @ B )
          = ( hAPP @ c_Complex_OIm @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(1052,axiom,
    class_RealVector_Oreal__normed__algebra @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra) ).

thf(4652,plain,
    class_RealVector_Oreal__normed__algebra @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(1244,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__natceiling__ge) ).

thf(4993,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1244]) ).

thf(1251,axiom,
    ( ( c_RComplete_Onatceiling @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__zero) ).

thf(5007,plain,
    ( ( c_RComplete_Onatceiling @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1251]) ).

thf(362,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
     => ( ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__pred) ).

thf(2329,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
     => ( ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(1068,axiom,
    class_RealVector_Oreal__normed__algebra__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__algebra__1) ).

thf(4684,plain,
    class_RealVector_Oreal__normed__algebra__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(511,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__idem) ).

thf(2842,plain,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(255,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oidom @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) )
      <=> ( ( B = A )
          | ( B
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_square__eq__iff) ).

thf(1952,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oidom @ C )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) )
         => ( ( B = A )
            | ( B
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) )
        & ( ( ( B = A )
            | ( B
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(245,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
       => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__neg) ).

thf(1926,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
       => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(736,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__cancel2) ).

thf(3627,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(278,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( ( B = A )
        | ( C
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__cancel1) ).

thf(2051,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( B = A )
          | ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( B = A )
          | ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(786,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__ge__minus__self) ).

thf(3806,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(1149,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Oadd) ).

thf(4827,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).

thf(423,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__imp__less__left) ).

thf(2535,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(116,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zless__mono) ).

thf(1496,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(1184,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__le) ).

thf(4890,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).

thf(809,axiom,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) )
      & ( ~ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ A )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zabs__def) ).

thf(3876,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ A ) ) )
      & ( ~ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(519,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__le__0__iff__le) ).

thf(2867,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(988,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Ozero) ).

thf(4473,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(490,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__Suc__right) ).

thf(2770,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(389,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Ozero) ).

thf(2418,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(1158,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_Odiff) ).

thf(4844,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).

thf(1151,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__minus__def) ).

thf(4832,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).

thf(1087,axiom,
    class_Groups_Oordered__comm__monoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__comm__monoid__add) ).

thf(4720,plain,
    class_Groups_Oordered__comm__monoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).

thf(217,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mult__cancel1) ).

thf(1841,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(396,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) ).

thf(2439,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(109,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__iff1) ).

thf(1474,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ E @ B ) ) @ D ) ) @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(909,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__le__imp__le) ).

thf(4232,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(474,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
       => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( c_Groups_Oone__class_Oone @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__pos) ).

thf(2725,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
       => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( c_Groups_Oone__class_Oone @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(142,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_BseqD) ).

thf(1594,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(592,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) @ ( hAPP @ c_Complex_ORe @ A ) ) ) ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_metric__bound__lemma) ).

thf(3110,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) @ ( hAPP @ c_Complex_ORe @ A ) ) ) ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ B ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(286,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Power_Opower @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Power_Opower) ).

thf(2086,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Power_Opower @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(606,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__comm__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__mult__strict__left__mono) ).

thf(3159,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__comm__semiring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(951,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__1__left) ).

thf(4362,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(749,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( ( C
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel__disj) ).

thf(3674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( B = A ) ) )
      & ( ( ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( B = A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(833,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C )
          = A )
      <=> ( B
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__imp__diff__is__add) ).

thf(3953,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C )
            = A )
         => ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) )
        & ( ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) )
         => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(498,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ E )
     => ( ( ( c_Groups_Ominus__class_Ominus @ E @ D @ C )
          = ( c_Groups_Ominus__class_Ominus @ E @ B @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
        <=> ( c_Orderings_Oord__class_Oless @ E @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__eq__diff__less) ).

thf(2797,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ E )
     => ( ( ( c_Groups_Ominus__class_Ominus @ E @ D @ C )
          = ( c_Groups_Ominus__class_Ominus @ E @ B @ A ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
           => ( c_Orderings_Oord__class_Oless @ E @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
           => ( c_Orderings_Oord__class_Oless @ E @ D @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(565,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__le__mono2) ).

thf(3016,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(194,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C @ E )
         => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ E )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_unique__quotient__lemma) ).

thf(1775,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C @ E )
         => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ E )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(438,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) @ ( c_Nat_OSuc @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq1) ).

thf(2592,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) @ ( c_Nat_OSuc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(110,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
      <=> ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ? [D: $i] :
              ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ A @ D ) ) ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__iff3) ).

thf(1477,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OBseq @ B @ A )
         => ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ? [D: $i] :
                ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ A @ D ) ) ) ) @ C ) ) )
        & ( ? [C: $i] :
              ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
              & ? [D: $i] :
                ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ A @ D ) ) ) ) @ C ) )
         => ( c_SEQ_OBseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(758,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_right__minus__eq) ).

thf(3706,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(656,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(3325,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(603,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I4_J) ).

thf(3142,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(57,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ B )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__antisym) ).

thf(1364,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ B )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(1223,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( A != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__not__refl2) ).

thf(4954,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1223]) ).

thf(552,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__self__eq__0) ).

thf(2967,plain,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(382,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__cancel__le) ).

thf(2390,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(561,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        & ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__is__0) ).

thf(2999,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(664,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( ( B
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        | ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__is__add) ).

thf(3348,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( ( B
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) )
      & ( ( ( ( B
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
       => ( ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(557,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).

thf(2984,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(205,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I12_J) ).

thf(1805,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(944,axiom,
    ! [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
    <=> ! [B: $i,C: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ C )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) @ ( hAPP @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_subseq__def) ).

thf(4331,plain,
    ! [A: $i] :
      ( ( ( c_SEQ_Osubseq @ A )
       => ! [B: $i,C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ C )
           => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) @ ( hAPP @ A @ C ) ) ) )
      & ( ! [B: $i,C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ C )
           => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) @ ( hAPP @ A @ C ) ) )
       => ( c_SEQ_Osubseq @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(971,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__diff__inverse) ).

thf(4426,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(1097,axiom,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_OIm,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_OisUCont) ).

thf(4741,plain,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_OIm,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).

thf(547,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__mult__distrib2) ).

thf(2952,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(646,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__semiring) ).

thf(3293,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(399,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__minus__cancel) ).

thf(2449,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(577,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ C )
        & ( class_RealVector_Oreal__normed__vector @ C ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Odiff) ).

thf(3048,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ C )
        & ( class_RealVector_Oreal__normed__vector @ C ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(41,axiom,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(1340,plain,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(1071,axiom,
    class_Groups_Olinordered__ab__group__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Olinordered__ab__group__add) ).

thf(4688,plain,
    class_Groups_Olinordered__ab__group__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(440,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__mono1) ).

thf(2597,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(965,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__diff__conv) ).

thf(4405,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) @ B ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) @ B )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(502,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__diff) ).

thf(2816,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(24,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
    <=> ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zless__le) ).

thf(1304,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
          & ( B != A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ A )
          & ( B != A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(615,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zless__add1__eq) ).

thf(3193,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
          | ( B = A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
          | ( B = A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(686,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__le__cancel__iff2) ).

thf(3427,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(828,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__comm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add) ).

thf(3936,plain,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__comm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(1194,axiom,
    class_Rings_Oring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__1) ).

thf(4901,plain,
    class_Rings_Oring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).

thf(722,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    <=> ( ( B
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        & ( A
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__eq__1__iff) ).

thf(3576,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( B
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
      & ( ( ( B
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(458,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__group__add) ).

thf(2665,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(1180,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__0) ).

thf(4886,plain,
    class_Rings_Ocomm__semiring__0 @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).

thf(1142,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ozero @ B )
     => ( ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Opsize @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ ( tc_Polynomial_Opoly @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_psize__eq__0__iff) ).

thf(4813,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ozero @ B )
     => ( ( ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Opsize @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ ( tc_Polynomial_Opoly @ B ) ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ ( tc_Polynomial_Opoly @ B ) ) )
         => ( ( c_Fundamental__Theorem__Algebra__Mirabelle_Opsize @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).

thf(559,axiom,
    ( ( c_Transcendental_Oln @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__one) ).

thf(2994,plain,
    ( ( c_Transcendental_Oln @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(1260,axiom,
    class_Groups_Oab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__semigroup__add) ).

thf(5025,plain,
    class_Groups_Oab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1260]) ).

thf(359,axiom,
    c_SEQ_Osubseq @ v_ga____,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_g_I1_J) ).

thf(2318,plain,
    c_SEQ_Osubseq @ v_ga____,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(676,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) ).

thf(3395,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(345,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ A )
     => ( class_Rings_Oring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1) ).

thf(2271,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ A )
     => ( class_Rings_Oring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(607,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__Suc1) ).

thf(3161,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(748,axiom,
    ! [A: $i,B: $i] :
      ( ( class_SEQ_Ocomplete__space @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
       => ( c_SEQ_Oconvergent @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Cauchy__convergent) ).

thf(3672,plain,
    ! [A: $i,B: $i] :
      ( ( class_SEQ_Ocomplete__space @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
       => ( c_SEQ_Oconvergent @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(1095,axiom,
    class_Groups_Omonoid__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__mult) ).

thf(4735,plain,
    class_Groups_Omonoid__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).

thf(840,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__mono) ).

thf(3981,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(414,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ouminus @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ouminus) ).

thf(2503,plain,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ouminus @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(892,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Odiff) ).

thf(4177,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) )
        = ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(450,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ E )
     => ( ( ( D != C )
          & ( B != A ) )
      <=> ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) )
         != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_crossproduct__noteq) ).

thf(2629,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ E )
     => ( ( ( ( D != C )
            & ( B != A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) )
           != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ B ) ) ) )
        & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) )
           != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ B ) ) )
         => ( ( D != C )
            & ( B != A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(540,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__diff__assoc2) ).

thf(2927,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) @ A )
        = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(838,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__0__right) ).

thf(3975,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(1141,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__mult) ).

thf(4812,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1141]) ).

thf(131,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__sgn__abs) ).

thf(1551,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(1240,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__add__def) ).

thf(4986,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ B ) ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1240]) ).

thf(890,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
      <=> ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__eq__add__iff1) ).

thf(4168,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B )
            = A ) )
        & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B )
            = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(674,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Ometric__space @ B )
     => ( ( c_SEQ_Oconvergent @ B @ A )
       => ( c_SEQ_OCauchy @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convergent__Cauchy) ).

thf(3391,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Ometric__space @ B )
     => ( ( c_SEQ_Oconvergent @ B @ A )
       => ( c_SEQ_OCauchy @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(739,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
      <=> ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
            & ( c_Orderings_Oord__class_Oless @ D @ C @ A ) )
          | ( ( c_Orderings_Oord__class_Oless @ D @ B @ ( c_Groups_Ozero__class_Ozero @ D ) )
            & ( c_Orderings_Oord__class_Oless @ D @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel__right__disj) ).

thf(3640,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
         => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
              & ( c_Orderings_Oord__class_Oless @ D @ C @ A ) )
            | ( ( c_Orderings_Oord__class_Oless @ D @ B @ ( c_Groups_Ozero__class_Ozero @ D ) )
              & ( c_Orderings_Oord__class_Oless @ D @ A @ C ) ) ) )
        & ( ( ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
              & ( c_Orderings_Oord__class_Oless @ D @ C @ A ) )
            | ( ( c_Orderings_Oord__class_Oless @ D @ B @ ( c_Groups_Ozero__class_Ozero @ D ) )
              & ( c_Orderings_Oord__class_Oless @ D @ A @ C ) ) )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(256,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__semigroup__add @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(1961,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__semigroup__add @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(593,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__strict__mono) ).

thf(3112,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(1060,axiom,
    class_Rings_Olinordered__idom @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__idom) ).

thf(4673,plain,
    class_Rings_Olinordered__idom @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(1220,axiom,
    class_Rings_Oring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring) ).

thf(4950,plain,
    class_Rings_Oring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1220]) ).

thf(796,axiom,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__add__one__not__less__self) ).

thf(3833,plain,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(625,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ A )
     => ( ( c_RealVector_Onorm__class_Onorm @ A @ ( c_Groups_Oone__class_Oone @ A ) )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__one) ).

thf(3228,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ A )
     => ( ( c_RealVector_Onorm__class_Onorm @ A @ ( c_Groups_Oone__class_Oone @ A ) )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(1229,axiom,
    class_Rings_Olinordered__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring) ).

thf(4971,plain,
    class_Rings_Olinordered__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1229]) ).

thf(1038,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4630,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).

thf(1256,axiom,
    class_Rings_Ocomm__ring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__ring) ).

thf(5018,plain,
    class_Rings_Ocomm__ring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1256]) ).

thf(507,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__less__mult) ).

thf(2831,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(843,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__gt__zero__cancel__iff) ).

thf(3988,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(418,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc2) ).

thf(2513,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(1103,axiom,
    class_Rings_Olinordered__ring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__ring) ).

thf(4749,plain,
    class_Rings_Olinordered__ring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).

thf(766,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ B ) ) @ A ) )
    <=> ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
        | ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__zpower__abs__iff) ).

thf(3737,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ B ) ) @ A ) )
       => ( ( B
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(193,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__0__le__add__iff) ).

thf(1770,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ A ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(1212,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( class_RealVector_Ometric__space @ D )
        & ( class_RealVector_Ometric__space @ C ) )
     => ( ( c_Lim_OisUCont @ C @ D @ B )
       => ( ( c_SEQ_OCauchy @ C @ A )
         => ( c_SEQ_OCauchy @ D @ ( c_COMBB @ C @ D @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_isUCont__Cauchy) ).

thf(4937,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( class_RealVector_Ometric__space @ D )
        & ( class_RealVector_Ometric__space @ C ) )
     => ( ( c_Lim_OisUCont @ C @ D @ B )
       => ( ( c_SEQ_OCauchy @ C @ A )
         => ( c_SEQ_OCauchy @ D @ ( c_COMBB @ C @ D @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1212]) ).

thf(377,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__1) ).

thf(2375,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(926,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__antisym) ).

thf(4272,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(1213,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( c_SEQ_Omonoseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monoseq__minus) ).

thf(4939,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( c_SEQ_Omonoseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ ( c_COMBB @ B @ B @ tc_Nat_Onat @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1213]) ).

thf(77,axiom,
    class_Groups_Oordered__ab__group__add__abs @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__group__add__abs) ).

thf(1400,plain,
    class_Groups_Oordered__ab__group__add__abs @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(266,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).

thf(2012,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(817,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ouminus @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ ( tc_fun @ C @ D ) ) @ B ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ D ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_uminus__apply) ).

thf(3904,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ouminus @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ ( tc_fun @ C @ D ) ) @ B ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ D ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(250,axiom,
    ( ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
    = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__of__real__minus__one) ).

thf(1941,plain,
    ( ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
    = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(899,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ C @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__minus__iff) ).

thf(4200,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(931,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__left__cancel) ).

thf(4287,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(986,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__increasing2) ).

thf(4469,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(196,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__add__Suc2) ).

thf(1780,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(503,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_i__mult__Complex) ).

thf(2821,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(1106,axiom,
    class_RealVector_Oreal__algebra__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__algebra__1) ).

thf(4754,plain,
    class_RealVector_Oreal__algebra__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).

thf(188,axiom,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) )
      & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__real__def) ).

thf(1753,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) )
      & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(1187,axiom,
    class_Groups_Omonoid__mult @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__mult) ).

thf(4894,plain,
    class_Groups_Omonoid__mult @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1187]) ).

thf(381,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( B = A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__less__eq__trans) ).

thf(2386,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( B = A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(778,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__less__imp__less) ).

thf(3786,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(1,conjecture,
    c_SEQ_OBseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ v_f____ ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ ( c_SEQ_OBseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ v_f____ ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(1267,plain,
    ~ ( c_SEQ_OBseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ v_f____ ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(975,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A )
          & ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_reals__Archimedean6) ).

thf(4437,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A )
          & ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(331,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__Suc__gt__zero) ).

thf(2224,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(791,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
        <=> ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel__left__pos) ).

thf(3818,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
       => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
           => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
           => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(1116,axiom,
    class_Rings_Olinordered__semiring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__1) ).

thf(4768,plain,
    class_Rings_Olinordered__semiring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).

thf(743,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( B = A )
      <=> ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__iff__diff__eq__0) ).

thf(3655,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( ( B = A )
         => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) )
        & ( ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

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

thf(1317,plain,
    class_Rings_Olinordered__idom @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(878,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__power__minus) ).

thf(4117,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
        = ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(584,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__add__le__0__iff) ).

thf(3080,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(481,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_Groups_Osgn__class_Osgn @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__minus) ).

thf(2747,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_Groups_Osgn__class_Osgn @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(289,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__strict) ).

thf(2092,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

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

thf(4092,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_RComplete_Onatfloor @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ B ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( c_RComplete_Onatfloor @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(1197,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__imp__le) ).

thf(4905,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).

thf(156,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
     => ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ? [C: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ C ) @ A ) ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_reduce__poly__simple) ).

thf(1637,plain,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
     => ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ? [C: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ C ) @ A ) ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(594,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( ( c_SEQ_Oconvergent @ C @ A )
         => ( c_SEQ_Oconvergent @ C @ ( c_COMBS @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convergent__add) ).

thf(3114,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( ( c_SEQ_Oconvergent @ C @ A )
         => ( c_SEQ_Oconvergent @ C @ ( c_COMBS @ tc_Nat_Onat @ C @ C @ ( c_COMBB @ C @ ( tc_fun @ C @ C ) @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(467,axiom,
    ! [A: $i] :
      ( ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Groups_Osgn__class_Osgn @ tc_Int_Oint @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      & ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_Int_Oint @ A )
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
          & ( ~ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_Int_Oint @ A )
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zsgn__def) ).

thf(2688,plain,
    ! [A: $i] :
      ( ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Groups_Osgn__class_Osgn @ tc_Int_Oint @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      & ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_Int_Oint @ A )
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
          & ( ~ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
           => ( ( c_Groups_Osgn__class_Osgn @ tc_Int_Oint @ A )
              = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(639,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Omonoid__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__mult) ).

thf(3273,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Omonoid__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(616,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) @ B )
          = ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__mult__pos) ).

thf(3202,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) @ B )
          = ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(225,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__add__one__gt__zero) ).

thf(1869,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(885,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Oone @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oone) ).

thf(4145,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Oone @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(365,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I9_J) ).

thf(2337,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(347,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B = A )
      <=> ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
          & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__eq__iff) ).

thf(2276,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( ( B = A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(1263,axiom,
    class_Rings_Ocomm__ring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring) ).

thf(5030,plain,
    class_Rings_Ocomm__ring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1263]) ).

thf(784,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__diff__left) ).

thf(3801,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(794,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__class_Odistrib) ).

thf(3828,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(882,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) )
          | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_split__mult__pos__le) ).

thf(4132,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A ) )
          | ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
            & ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(164,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
      <=> ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__iff__diff__le__0) ).

thf(1658,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(1249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ! [E: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ E )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ ( hAPP @ B @ E ) ) @ A ) )
       => ( c_SEQ_OBseq @ D @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_BseqI2_H) ).

thf(5001,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ! [E: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ E )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ ( hAPP @ B @ E ) ) @ A ) )
       => ( c_SEQ_OBseq @ D @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1249]) ).

thf(469,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    <=> ( A
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__add__eq__0__iff) ).

thf(2701,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( A
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) )
      & ( ( A
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(913,axiom,
    ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ c_Complex_Oii )
    = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_i__mult__eq2) ).

thf(4241,plain,
    ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) @ c_Complex_Oii )
    = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(441,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
        | ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__zero__less__power__iff) ).

thf(2599,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(1130,axiom,
    class_Groups_Ogroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ogroup__add) ).

thf(4800,plain,
    class_Groups_Ogroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).

thf(1238,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ B )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__le__antisym) ).

thf(4982,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ B )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1238]) ).

thf(170,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left__mono) ).

thf(1679,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(681,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).

thf(3412,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(574,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Opos__bounded) ).

thf(3041,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(802,axiom,
    ! [A: $i,B: $i] :
      ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__mult) ).

thf(3853,plain,
    ! [A: $i,B: $i] :
      ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(39,axiom,
    class_Rings_Oordered__ring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring) ).

thf(1336,plain,
    class_Rings_Oordered__ring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(1177,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__add) ).

thf(4875,plain,
    class_Groups_Ocomm__monoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1177]) ).

thf(760,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__antisym__conv) ).

thf(3715,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
           => ( A = B ) )
          & ( ( A = B )
           => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(372,axiom,
    ! [A: $i,B: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__add__less1) ).

thf(2354,plain,
    ! [A: $i,B: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(54,axiom,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint )
   != ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__0__neq__1) ).

thf(1358,plain,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint )
   != ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(701,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__minus__mult) ).

thf(3488,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(1132,axiom,
    class_Rings_Osemiring__0 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring__0) ).

thf(4802,plain,
    class_Rings_Osemiring__0 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).

thf(1225,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_RealDef_Oreal @ tc_Nat_Onat @ B )
        = ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__inject) ).

thf(4959,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_RealDef_Oreal @ tc_Nat_Onat @ B )
          = ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( c_RealDef_Oreal @ tc_Nat_Onat @ B )
          = ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1225]) ).

thf(682,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1) ).

thf(3415,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(14,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__zero__iff) ).

thf(1284,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(373,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Limits_Ominus__diff__minus) ).

thf(2357,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(433,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__le__power) ).

thf(2579,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oone__class_Oone @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oone__class_Oone @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(119,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ B ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__zpower__abs) ).

thf(1504,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Int_Oint ) @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(1041,axiom,
    class_Groups_Oabs__if @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oabs__if) ).

thf(4634,plain,
    class_Groups_Oabs__if @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).

thf(568,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__antisym__conv3) ).

thf(3024,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( ( ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B )
           => ( A = B ) )
          & ( ( A = B )
           => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(473,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__1__right) ).

thf(2722,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(666,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__mono1) ).

thf(3366,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(920,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_xt1_I6_J) ).

thf(4257,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ A @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(545,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__power) ).

thf(2948,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(754,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__not__less__zero) ).

thf(3697,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(598,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add) ).

thf(3124,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(392,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__eq__le__trans) ).

thf(2426,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(204,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__add__abs) ).

thf(1802,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( c_Groups_Oabs__class_Oabs @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(703,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
    <=> ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__less__def) ).

thf(3493,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
          & ( B != A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
          & ( B != A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(1036,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring__1) ).

thf(4624,plain,
    class_Rings_Ocomm__semiring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).

thf(1027,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__semigroup__add) ).

thf(4611,plain,
    class_Groups_Ocancel__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).

thf(1234,axiom,
    class_RealVector_Oreal__normed__algebra__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1) ).

thf(4976,plain,
    class_RealVector_Oreal__normed__algebra__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1234]) ).

thf(112,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__of__real__add__Complex) ).

thf(1485,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(876,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Osgn__if @ B )
     => ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
             => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
                = ( c_Groups_Oone__class_Oone @ B ) ) )
            & ( ~ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
             => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
                = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__if) ).

thf(4106,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Osgn__if @ B )
     => ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
             => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
                = ( c_Groups_Oone__class_Oone @ B ) ) )
            & ( ~ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
             => ( ( c_Groups_Osgn__class_Osgn @ B @ A )
                = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(1243,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__trans) ).

thf(4991,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1243]) ).

thf(120,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oring @ F )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
      <=> ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_eq__add__iff2) ).

thf(1506,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oring @ F )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
         => ( C
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) ) )
        & ( ( C
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(861,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__power__abs) ).

thf(4046,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(1210,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__minus) ).

thf(4932,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1210]) ).

thf(475,axiom,
    ! [A: $i,B: $i] :
      ( ( B
        = ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) ) )
     => ( ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__power) ).

thf(2728,plain,
    ! [A: $i,B: $i] :
      ( ( B
        = ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) ) )
     => ( ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(924,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ B )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__minus) ).

thf(4267,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ B )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(272,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__ge__zero__iff) ).

thf(2030,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(61,axiom,
    class_Orderings_Olinorder @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Olinorder) ).

thf(1372,plain,
    class_Orderings_Olinorder @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(1155,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OBseq @ tc_RealDef_Oreal @ A )
     => ( ! [B: $i,C: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ C )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ A @ B ) @ ( hAPP @ A @ C ) ) )
       => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__mono__convergent) ).

thf(4838,plain,
    ! [A: $i] :
      ( ( c_SEQ_OBseq @ tc_RealDef_Oreal @ A )
     => ( ! [B: $i,C: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ C )
           => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ A @ B ) @ ( hAPP @ A @ C ) ) )
       => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).

thf(634,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
    <=> ( C = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__right__cancel) ).

thf(3256,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( C = A ) )
      & ( ( C = A )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(528,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__le__imp__le__exp) ).

thf(2892,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(7,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__assoc) ).

thf(1272,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(605,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__minus__self__iff) ).

thf(3156,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(1127,axiom,
    class_RealVector_Oreal__normed__vector @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__vector) ).

thf(4795,plain,
    class_RealVector_Oreal__normed__vector @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).

thf(1143,axiom,
    class_Rings_Oordered__comm__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__comm__semiring) ).

thf(4819,plain,
    class_Rings_Oordered__comm__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).

thf(750,axiom,
    ! [A: $i,B: $i] :
      ( ( c_SEQ_Osubseq @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_seq__suble) ).

thf(3684,plain,
    ! [A: $i,B: $i] :
      ( ( c_SEQ_Osubseq @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(644,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__less__self) ).

thf(3288,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(539,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__0__right) ).

thf(2924,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(928,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_RComplete_Onatceiling @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatceiling @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__add) ).

thf(4278,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_RComplete_Onatceiling @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatceiling @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(402,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_SEQ_Ocomplete__space @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( ( c_SEQ_Osubseq @ A )
         => ( c_SEQ_Oconvergent @ C @ ( c_Fun_Ocomp @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_convergent__subseq__convergent) ).

thf(2458,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_SEQ_Ocomplete__space @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( ( c_SEQ_Osubseq @ A )
         => ( c_SEQ_Oconvergent @ C @ ( c_Fun_Ocomp @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(1163,axiom,
    class_Rings_Ocomm__ring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__ring__1) ).

thf(4851,plain,
    class_Rings_Ocomm__ring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).

thf(582,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__eq__diff__pred) ).

thf(3067,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Nat_OSuc @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(209,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Rings_Ocomm__semiring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__1) ).

thf(1817,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Rings_Ocomm__semiring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(808,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__cancel__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(3874,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__cancel__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

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

thf(2194,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(162,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Oone__class_Oone @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__increasing) ).

thf(1654,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Oone__class_Oone @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(326,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) @ B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__le__self) ).

thf(2209,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(1049,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__refl) ).

thf(4646,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(319,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__strict__left__mono__neg) ).

thf(2187,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ A @ ( c_Groups_Ozero__class_Ozero @ D ) )
         => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(518,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_LIMSEQ__inverse__realpow__zero__lemma) ).

thf(2865,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(522,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__eq__less__trans) ).

thf(2875,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( c_Orderings_Oord__class_Oless @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(395,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ F )
     => ( ( E
         != ( c_Groups_Ozero__class_Ozero @ F ) )
       => ( ( ( D = C )
            & ( B != A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ B ) )
           != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__scale__eq__noteq) ).

thf(2435,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ F )
     => ( ( E
         != ( c_Groups_Ozero__class_Ozero @ F ) )
       => ( ( ( D = C )
            & ( B != A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ B ) )
           != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(1044,axiom,
    class_Groups_Osgn__if @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Osgn__if) ).

thf(4639,plain,
    class_Groups_Osgn__if @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(216,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Oone__class_Oone @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__Suc__less) ).

thf(1839,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Oone__class_Oone @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(1051,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocomm__monoid__mult) ).

thf(4651,plain,
    class_Groups_Ocomm__monoid__mult @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).

thf(884,axiom,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) )
      & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__abs__def) ).

thf(4138,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) )
      & ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(26,axiom,
    class_Groups_Oab__group__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__group__add) ).

thf(1316,plain,
    class_Groups_Oab__group__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(1074,axiom,
    class_Groups_Oone @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oone) ).

thf(4698,plain,
    class_Groups_Oone @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(1090,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__semigroup__add) ).

thf(4727,plain,
    class_Groups_Ocancel__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).

thf(287,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_termination__basic__simps_I3_J) ).

thf(2088,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(942,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).

thf(4325,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(1019,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re) ).

thf(4587,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).

thf(1118,axiom,
    class_Power_Opower @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Power_Opower) ).

thf(4770,plain,
    class_Power_Opower @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).

thf(258,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__assoc) ).

thf(1966,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(254,axiom,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__zeroE) ).

thf(1949,plain,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(1112,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im_OCauchy) ).

thf(4763,plain,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).

thf(520,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonpos__nonpos) ).

thf(2870,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(1217,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__commute) ).

thf(4944,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1217]) ).

thf(1175,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult) ).

thf(4873,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).

thf(31,axiom,
    class_Orderings_Oord @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Oord) ).

thf(1321,plain,
    class_Orderings_Oord @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(214,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_left__minus) ).

thf(1834,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(123,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( B = A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__eq__refl) ).

thf(1517,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( B = A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(1125,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
        | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__neq__iff) ).

thf(4780,plain,
    ! [A: $i,B: $i] :
      ( ( ( B != A )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
          | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
          | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ B ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).

thf(443,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__lessD1) ).

thf(2611,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(894,axiom,
    ! [A: $i] :
      ( ( class_Groups_Osgn__if @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn0) ).

thf(4185,plain,
    ! [A: $i] :
      ( ( class_Groups_Osgn__if @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(580,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( B != A )
      <=> ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
          | ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__neq__iff) ).

thf(3056,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ( B != A )
         => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
            | ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
            | ( c_Orderings_Oord__class_Oless @ C @ A @ B ) )
         => ( B != A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(332,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_SEQ_OBseq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__realpow) ).

thf(2226,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_SEQ_OBseq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(103,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ A )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__one__le__iff__zero__less) ).

thf(1453,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(283,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__diff__inverse) ).

thf(2071,plain,
    ! [A: $i,B: $i] :
      ( ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(456,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__mult__le__cancel1) ).

thf(2658,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(492,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ab__diff__minus) ).

thf(2775,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(534,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__of__real__mult__Complex) ).

thf(2907,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(936,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__right) ).

thf(4310,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(496,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__minus__le__zero) ).

thf(2792,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(1064,axiom,
    class_Groups_Ocomm__monoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocomm__monoid__add) ).

thf(4678,plain,
    class_Groups_Ocomm__monoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).

thf(573,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__mono) ).

thf(3039,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(9,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oord @ B )
     => ( class_Orderings_Oord @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Orderings_Oord) ).

thf(1276,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oord @ B )
     => ( class_Orderings_Oord @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(36,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_compl__eq__compl__iff) ).

thf(1326,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(1082,axiom,
    class_Rings_Ozero__neq__one @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ozero__neq__one) ).

thf(4710,plain,
    class_Rings_Ozero__neq__one @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).

thf(346,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ A ) @ ( c_Groups_Oone__class_Oone @ C ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) ).

thf(2273,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ A ) @ ( c_Groups_Oone__class_Oone @ C ) ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(1136,axiom,
    class_Rings_Ocomm__ring__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring__1) ).

thf(4806,plain,
    class_Rings_Ocomm__ring__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).

thf(42,axiom,
    class_Orderings_Opreorder @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Opreorder) ).

thf(1341,plain,
    class_Orderings_Opreorder @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(428,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ F @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add__iff2) ).

thf(2554,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ F @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ F @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( c_Groups_Ominus__class_Ominus @ F @ B @ E ) ) @ D ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ F @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ E ) @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ D ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(1067,axiom,
    class_SEQ_Ocomplete__space @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__SEQ_Ocomplete__space) ).

thf(4683,plain,
    class_SEQ_Ocomplete__space @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(917,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4250,plain,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(500,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__less__imp__less) ).

thf(2811,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ B )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(945,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__imp__diff__less) ).

thf(4338,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(442,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__add) ).

thf(2608,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(211,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__less__imp__less) ).

thf(1822,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(732,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im__complex__of__real) ).

thf(3613,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(1200,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__comm__monoid__add) ).

thf(4913,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).

thf(168,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__mult__minus) ).

thf(1674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oring @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(515,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__add__less) ).

thf(2855,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(514,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__trans) ).

thf(2853,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(793,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ C )
        & ( class_RealVector_Oreal__normed__vector @ C ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Oadd) ).

thf(3825,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ C )
        & ( class_RealVector_Oreal__normed__vector @ C ) )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(767,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__mono_H) ).

thf(3748,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( c_Orderings_Oord__class_Oless__eq @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(271,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ C ) @ B )
              = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_realpow__pos__nth) ).

thf(2026,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ? [C: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
            & ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ C ) @ B )
              = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(709,axiom,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__less0) ).

thf(3516,plain,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(994,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) )
       => ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_rabs__ratiotest__lemma) ).

thf(4495,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ) )
       => ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(125,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__nonpos__nonpos) ).

thf(1523,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(1079,axiom,
    class_RealVector_Oreal__normed__div__algebra @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra) ).

thf(4705,plain,
    class_RealVector_Oreal__normed__div__algebra @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).

thf(804,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__antisym__conv1) ).

thf(3859,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
           => ( B = A ) )
          & ( ( B = A )
           => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(602,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
        & ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_one__le__mult__iff) ).

thf(3136,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
          & ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B )
          & ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(667,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).

thf(3368,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(999,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__less__add__iff1) ).

thf(4527,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E @ D )
     => ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ D @ E ) ) @ C ) ) @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(388,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( c_SEQ_Oconvergent @ C @ ( c_COMBB @ C @ C @ tc_Nat_Onat @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right_Oconvergent) ).

thf(2416,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( c_SEQ_Oconvergent @ C @ B )
       => ( c_SEQ_Oconvergent @ C @ ( c_COMBB @ C @ C @ tc_Nat_Onat @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(1193,axiom,
    class_Groups_Ouminus @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ouminus) ).

thf(4900,plain,
    class_Groups_Ouminus @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).

thf(306,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
     => ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( A
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__lt__two__imp__zero__or__one) ).

thf(2152,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
     => ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( A
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(980,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__nat_Odiff__0) ).

thf(4451,plain,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(585,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_double__add__le__zero__iff__single__add__le__zero) ).

thf(3085,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(1100,axiom,
    class_Groups_Ominus @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ominus) ).

thf(4745,plain,
    class_Groups_Ominus @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).

thf(265,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) ).

thf(2009,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(898,axiom,
    ! [A: $i] :
      ( ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        | ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ A @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        | ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ A ) @ c_Complex_Oii ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        | ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ A @ c_Complex_Oii ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_unimodular__reduce__norm) ).

thf(4197,plain,
    ! [A: $i] :
      ( ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        | ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ A @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        | ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ A ) @ c_Complex_Oii ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        | ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ A @ c_Complex_Oii ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(536,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__add__commute) ).

thf(2912,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(887,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( hAPP @ B @ D ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( hAPP @ B @ C ) )
         => ? [D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_int__val__lemma) ).

thf(4153,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oabs__class_Oabs @ tc_Int_Oint @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( hAPP @ B @ D ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ ( hAPP @ B @ C ) )
         => ? [D: $i] :
              ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D @ C )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(1109,axiom,
    class_Groups_Oab__group__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__group__add) ).

thf(4759,plain,
    class_Groups_Oab__group__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).

thf(714,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
    <=> ( ( B
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        & ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__eq__1__iff) ).

thf(3529,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
       => ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(310,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__mod__minus__le__complex__mod) ).

thf(2162,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(781,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
        <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__diff__iff) ).

thf(3793,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ A )
       => ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) )
           => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) )
          & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
           => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

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

thf(1607,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
       => ( ( B != A )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(844,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__mult) ).

thf(3993,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ C )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ B ) ) @ ( hAPP @ ( c_RealVector_Oof__real @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(165,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__comm__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__comm__semiring) ).

thf(1661,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__comm__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(351,axiom,
    ! [A: $i,B: $i] :
      ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__power) ).

thf(2294,plain,
    ! [A: $i,B: $i] :
      ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(358,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Im) ).

thf(2315,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(390,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__def) ).

thf(2421,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(173,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zadd__zmult__distrib2) ).

thf(1689,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(839,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ A )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__0) ).

thf(3978,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ A )
     => ( ( hAPP @ ( c_RealVector_Oof__real @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(613,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( B = A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ord__le__eq__trans) ).

thf(3181,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ C @ B )
       => ( ( B = A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(407,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__inverse) ).

thf(2475,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(1165,axiom,
    class_Groups_Ozero @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ozero) ).

thf(4855,plain,
    class_Groups_Ozero @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1165]) ).

thf(753,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__less) ).

thf(3694,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ A @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(723,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ C @ B ) ) @ A )
      = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib) ).

thf(3585,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ C @ B ) ) @ A )
      = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(687,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__neg__pos) ).

thf(3430,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(195,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).

thf(1777,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(1122,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Nat_Osize__class_Osize @ tc_Complex_Ocomplex @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex_Osize_I2_J) ).

thf(4774,plain,
    ! [A: $i,B: $i] :
      ( ( c_Nat_Osize__class_Osize @ tc_Complex_Ocomplex @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).

thf(17,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ouminus @ B )
     => ( class_Groups_Ouminus @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun__Groups_Ouminus) ).

thf(1294,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ouminus @ B )
     => ( class_Groups_Ouminus @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(404,axiom,
    ! [A: $i] :
      ( ( c_Complex_Orcis @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
      = ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_rcis__zero__arg) ).

thf(2462,plain,
    ! [A: $i] :
      ( ( c_Complex_Orcis @ A @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
      = ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(959,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ ( c_Groups_Ozero__class_Ozero @ D ) )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
        <=> ( c_Orderings_Oord__class_Oless @ D @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__cancel__left__neg) ).

thf(4387,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ ( c_Groups_Ozero__class_Ozero @ D ) )
       => ( ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) )
           => ( c_Orderings_Oord__class_Oless @ D @ A @ B ) )
          & ( ( c_Orderings_Oord__class_Oless @ D @ A @ B )
           => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(281,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Odecseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_decseq__imp__monoseq) ).

thf(2066,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Odecseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(868,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__triangle__ineq2) ).

thf(4074,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(978,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__imp__le) ).

thf(4446,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(479,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
      <=> ( A
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__eq__0__iff) ).

thf(2739,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( A
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) )
        & ( ( A
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(412,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Oring__1 @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) @ ( c_Groups_Ominus__class_Ominus @ B @ A @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__squared__diff__one__factored) ).

thf(2494,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Oring__1 @ B )
     => ( ( c_Groups_Ominus__class_Ominus @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) ) ) @ ( c_Groups_Ominus__class_Ominus @ B @ A @ ( c_Groups_Oone__class_Oone @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(100,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ozero__left) ).

thf(1444,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(1172,axiom,
    ! [A: $i,B: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ B )
        & ( class_RealVector_Oreal__normed__vector @ B ) )
     => ( ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ A )
       => ( c_SEQ_OCauchy @ B @ ( c_COMBB @ tc_RealDef_Oreal @ B @ tc_Nat_Onat @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_OCauchy) ).

thf(4869,plain,
    ! [A: $i,B: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ B )
        & ( class_RealVector_Oreal__normed__vector @ B ) )
     => ( ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ A )
       => ( c_SEQ_OCauchy @ B @ ( c_COMBB @ tc_RealDef_Oreal @ B @ tc_Nat_Onat @ ( c_RealVector_Oof__real @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1172]) ).

thf(538,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( ( c_Groups_Ominus__class_Ominus @ E @ D @ C )
          = ( c_Groups_Ominus__class_Ominus @ E @ B @ A ) )
       => ( ( D = C )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__eq__diff__eq) ).

thf(2917,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( ( c_Groups_Ominus__class_Ominus @ E @ D @ C )
          = ( c_Groups_Ominus__class_Ominus @ E @ B @ A ) )
       => ( ( ( D = C )
           => ( B = A ) )
          & ( ( B = A )
           => ( D = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(955,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
        | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__gr__0) ).

thf(4372,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B )
          | ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(671,axiom,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__add1) ).

thf(3379,plain,
    ! [A: $i,B: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(541,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__square) ).

thf(2930,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(858,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__ge__zero) ).

thf(4039,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(121,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) )
        = ( c_RealVector_Onorm__class_Onorm @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__norm__cancel) ).

thf(1512,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ A ) )
        = ( c_RealVector_Onorm__class_Onorm @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(1252,axiom,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__irrefl__nat) ).

thf(5009,plain,
    ! [A: $i] :
      ~ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1252]) ).

thf(12,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__0) ).

thf(1280,plain,
    class_Rings_Ocomm__semiring__0 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(818,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__1__right) ).

thf(3907,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(153,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__imp__not__less) ).

thf(1625,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
       => ~ ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(421,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__le__zero__iff) ).

thf(2522,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(780,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__Suc__1) ).

thf(3790,plain,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(953,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zless__imp__add1__zle) ).

thf(4367,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(158,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( B != A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__neq__implies__less) ).

thf(1642,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( B != A )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

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

thf(2102,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(1204,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Oadd) ).

thf(4917,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_ORe @ B ) ) @ ( hAPP @ c_Complex_ORe @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1204]) ).

thf(877,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__add) ).

thf(4114,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(963,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( B != A )
       => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_linorder__neqE) ).

thf(4399,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( B != A )
       => ( ~ ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(1080,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
      | ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__le__linear) ).

thf(4706,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
      | ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).

thf(707,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Oone__class_Oone @ C ) )
           => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__right__le__one__le) ).

thf(3511,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Oone__class_Oone @ C ) )
           => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(307,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__comm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__comm__monoid__add) ).

thf(2155,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__comm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(341,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__gt__zero__imp__gt__one) ).

thf(2255,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ A ) )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(805,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( c_Fun_Ocomp @ H @ G @ F @ E @ D )
        = ( c_Fun_Ocomp @ C @ G @ F @ B @ A ) )
     => ! [I: $i] :
          ( ( hAPP @ E @ ( hAPP @ D @ I ) )
          = ( hAPP @ B @ ( hAPP @ A @ I ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_o__eq__elim) ).

thf(3865,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( c_Fun_Ocomp @ H @ G @ F @ E @ D )
        = ( c_Fun_Ocomp @ C @ G @ F @ B @ A ) )
     => ! [I: $i] :
          ( ( hAPP @ E @ ( hAPP @ D @ I ) )
          = ( hAPP @ B @ ( hAPP @ A @ I ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(472,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oring @ E )
     => ( ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__diff__mult) ).

thf(2719,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oring @ E )
     => ( ( c_Groups_Ominus__class_Ominus @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ ( c_Groups_Ominus__class_Ominus @ E @ C @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Ominus__class_Ominus @ E @ D @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(919,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ogroup__add @ A )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ A ) @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__zero) ).

thf(4254,plain,
    ! [A: $i] :
      ( ( class_Groups_Ogroup__add @ A )
     => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ A ) @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(1250,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Complex_Ocomplex_Ocomplex__size @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex_Osize_I1_J) ).

thf(5004,plain,
    ! [A: $i,B: $i] :
      ( ( c_Complex_Ocomplex_Ocomplex__size @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1250]) ).

thf(1140,axiom,
    class_Power_Opower @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Power_Opower) ).

thf(4811,plain,
    class_Power_Opower @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1140]) ).

thf(436,axiom,
    ! [A: $i] :
      ( ( c_Complex_Ocomplex_OComplex @ ( hAPP @ c_Complex_ORe @ A ) @ ( hAPP @ c_Complex_OIm @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__surj) ).

thf(2587,plain,
    ! [A: $i] :
      ( ( c_Complex_Ocomplex_OComplex @ ( hAPP @ c_Complex_ORe @ A ) @ ( hAPP @ c_Complex_OIm @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(1023,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__Suc0) ).

thf(4596,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).

thf(813,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) ).

thf(3892,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(694,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( ( c_Groups_Ozero__class_Ozero @ B )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
      <=> ( ( c_Groups_Ozero__class_Ozero @ B )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__0__equal__iff__equal) ).

thf(3464,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ogroup__add @ B )
     => ( ( ( ( c_Groups_Ozero__class_Ozero @ B )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
         => ( ( c_Groups_Ozero__class_Ozero @ B )
            = A ) )
        & ( ( ( c_Groups_Ozero__class_Ozero @ B )
            = A )
         => ( ( c_Groups_Ozero__class_Ozero @ B )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(66,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_lessI) ).

thf(1380,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A @ ( c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(711,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_trans__less__add2) ).

thf(3521,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower_Opower @ D @ C @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = C ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power_Opower_Opower__0) ).

thf(1876,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower_Opower @ D @ C @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(23,axiom,
    class_Rings_Oordered__comm__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__comm__semiring) ).

thf(1303,plain,
    class_Rings_Oordered__comm__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(1156,axiom,
    class_RealVector_Oreal__normed__vector @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__vector) ).

thf(4841,plain,
    class_RealVector_Oreal__normed__vector @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).

thf(1016,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ D @ C ) ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_decr__mult__lemma) ).

thf(4579,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ C )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ D @ C ) ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(1179,axiom,
    class_Rings_Olinordered__ring__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__ring__strict) ).

thf(4885,plain,
    class_Rings_Olinordered__ring__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1179]) ).

thf(489,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Osgn__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Osgn__if) ).

thf(2768,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Osgn__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(221,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( class_RealVector_Oreal__normed__vector @ D )
        & ( class_Orderings_Oord @ C ) )
     => ( ! [E: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ E )
           => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ A @ B ) @ ( hAPP @ A @ E ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ! [E: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ E )
           => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_RealVector_Onorm__class_Onorm @ D @ ( hAPP @ A @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_lemmaCauchy) ).

thf(1859,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( class_RealVector_Oreal__normed__vector @ D )
        & ( class_Orderings_Oord @ C ) )
     => ( ! [E: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ E )
           => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ ( c_Groups_Ominus__class_Ominus @ D @ ( hAPP @ A @ B ) @ ( hAPP @ A @ E ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ! [E: $i] :
            ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ E )
           => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ ( hAPP @ A @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ ( c_RealVector_Onorm__class_Onorm @ D @ ( hAPP @ A @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(657,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__ring) ).

thf(3328,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(56,axiom,
    class_Rings_Oring__1__no__zero__divisors @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1__no__zero__divisors) ).

thf(1363,plain,
    class_Rings_Oring__1__no__zero__divisors @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(1228,axiom,
    ! [A: $i] :
      ( ( c_Complex_Orcis @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_rcis__zero__mod) ).

thf(4968,plain,
    ! [A: $i] :
      ( ( c_Complex_Orcis @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1228]) ).

thf(801,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__add__0) ).

thf(3850,plain,
    ! [A: $i,B: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(698,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__double__add__iff__zero__less__single__add) ).

thf(3479,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(200,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__triangle__ineq) ).

thf(1789,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(375,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__sgn) ).

thf(2366,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) )
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(178,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Complex_Ocomplex_OComplex @ C @ B )
        = ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
    <=> ( ( C = A )
        & ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__eq__cancel__iff2) ).

thf(1710,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( c_Complex_Ocomplex_OComplex @ C @ B )
          = ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) )
       => ( ( C = A )
          & ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) )
      & ( ( ( C = A )
          & ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
       => ( ( c_Complex_Ocomplex_OComplex @ C @ B )
          = ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(471,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__diff__conv) ).

thf(2714,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(446,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__nonneg__nonneg) ).

thf(2619,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(800,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__cancel1) ).

thf(3843,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
         => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(939,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__less__imp__less__exp) ).

thf(4318,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(323,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ B ) ) @ A )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__zminus) ).

thf(2201,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ B ) ) @ A )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(1010,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__pos__nonneg) ).

thf(4565,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).

thf(1123,axiom,
    class_Groups_Ouminus @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ouminus) ).

thf(4777,plain,
    class_Groups_Ouminus @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).

thf(1107,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__1) ).

thf(4755,plain,
    class_Rings_Ocomm__semiring__1 @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).

thf(364,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__one__right) ).

thf(2334,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(1188,axiom,
    class_Rings_Ozero__neq__one @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ozero__neq__one) ).

thf(4895,plain,
    class_Rings_Ozero__neq__one @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).

thf(929,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) ) @ c_Complex_Oii )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__of__real__i) ).

thf(4281,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( hAPP @ ( c_RealVector_Oof__real @ tc_Complex_Ocomplex ) @ A ) ) @ c_Complex_Oii )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(1039,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OBseq @ C @ B )
       => ( c_SEQ_OBseq @ C @ ( c_COMBB @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Bseq__ignore__initial__segment) ).

thf(4631,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OBseq @ C @ B )
       => ( c_SEQ_OBseq @ C @ ( c_COMBB @ tc_Nat_Onat @ C @ tc_Nat_Onat @ B @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(785,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__le__mono) ).

thf(3804,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(38,axiom,
    class_Rings_Ozero__neq__one @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ozero__neq__one) ).

thf(1335,plain,
    class_Rings_Ozero__neq__one @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(1257,axiom,
    ! [A: $i] :
      ( ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Oconvergent) ).

thf(5019,plain,
    ! [A: $i] :
      ( ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_ORe @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1257]) ).

thf(871,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ E )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ C ) ) )
      <=> ( ( D = B )
          | ( C = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_crossproduct__eq) ).

thf(4083,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ E )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ C ) ) )
         => ( ( D = B )
            | ( C = A ) ) )
        & ( ( ( D = B )
            | ( C = A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(853,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ogroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ogroup__add) ).

thf(4024,plain,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ogroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(1169,axiom,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(4865,plain,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).

thf(569,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__sgn__pos) ).

thf(3030,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ tc_RealDef_Oreal @ A )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(774,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
    <=> ( ( ( hAPP @ c_Complex_ORe @ B )
          = ( hAPP @ c_Complex_ORe @ A ) )
        & ( ( hAPP @ c_Complex_OIm @ B )
          = ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_expand__complex__eq) ).

thf(3765,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
       => ( ( ( hAPP @ c_Complex_ORe @ B )
            = ( hAPP @ c_Complex_ORe @ A ) )
          & ( ( hAPP @ c_Complex_OIm @ B )
            = ( hAPP @ c_Complex_OIm @ A ) ) ) )
      & ( ( ( ( hAPP @ c_Complex_ORe @ B )
            = ( hAPP @ c_Complex_ORe @ A ) )
          & ( ( hAPP @ c_Complex_OIm @ B )
            = ( hAPP @ c_Complex_OIm @ A ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(280,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ A )
      <=> ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__less__nonneg) ).

thf(2063,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ A )
         => ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(148,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
       => ( ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ A @ D ) ) @ B )
         => ( c_SEQ_OBseq @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_BseqI) ).

thf(1610,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
       => ( ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ A @ D ) ) @ B )
         => ( c_SEQ_OBseq @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(1207,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) @ ( hAPP @ c_Complex_ORe @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Re_Odiff) ).

thf(4923,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ B ) @ ( hAPP @ c_Complex_ORe @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1207]) ).

thf(1160,axiom,
    class_RealVector_Oreal__algebra__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__algebra__1) ).

thf(4848,plain,
    class_RealVector_Oreal__algebra__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).

thf(513,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__mult__right__cancel) ).

thf(2847,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(806,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__idom @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Oabs__class_Oabs @ E @ D ) ) @ ( c_Groups_Oabs__class_Oabs @ E @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__mult__less) ).

thf(3869,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__idom @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( c_Groups_Oabs__class_Oabs @ E @ D ) ) @ ( c_Groups_Oabs__class_Oabs @ E @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(1178,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Complex_Ocomplex_OComplex @ B @ A )
        = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    <=> ( ( B
          = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
        & ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Complex__eq__1) ).

thf(4876,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Complex_Ocomplex_OComplex @ B @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
       => ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) )
      & ( ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
       => ( ( c_Complex_Ocomplex_OComplex @ B @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).

thf(966,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_plus__nat_Oadd__0) ).

thf(4410,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(237,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__diff1__eq) ).

thf(1902,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B @ A )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(468,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Ozero__class_Ozero @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ A ) @ ( c_Groups_Oone__class_Oone @ A ) ) @ ( c_Groups_Oone__class_Oone @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__less__two) ).

thf(2699,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Ozero__class_Ozero @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ A ) @ ( c_Groups_Oone__class_Oone @ A ) ) @ ( c_Groups_Oone__class_Oone @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(996,axiom,
    ! [A: $i] :
      ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_not__real__square__gt__zero) ).

thf(4501,plain,
    ! [A: $i] :
      ( ( ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ~ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(376,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ ( c_RealVector_Onorm__class_Onorm @ D @ A ) ) )
         => ( B
            = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__ratiotest__lemma) ).

thf(2372,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ ( c_RealVector_Onorm__class_Onorm @ D @ A ) ) )
         => ( B
            = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(309,axiom,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( c_RealVector_Oof__real @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ C ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real_Ononneg__bounded) ).

thf(2159,plain,
    ! [A: $i] :
      ( ( ( class_RealVector_Oreal__algebra__1 @ A )
        & ( class_RealVector_Oreal__normed__vector @ A ) )
     => ? [B: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
          & ! [C: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ A @ ( hAPP @ ( c_RealVector_Oof__real @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ C ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(476,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
      <=> ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__less__zero__iff) ).

thf(2731,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(881,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).

thf(4129,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ A ) @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(260,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( c_Lim_OisUCont @ B @ B @ ( c_COMBC @ B @ B @ B @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_OisUCont) ).

thf(1977,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ( c_Lim_OisUCont @ B @ B @ ( c_COMBC @ B @ B @ B @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(393,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__less__cancel1) ).

thf(2430,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) )
        & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(3,axiom,
    class_Rings_Oordered__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__semiring) ).

thf(1268,plain,
    class_Rings_Oordered__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(756,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_monoseq__realpow) ).

thf(3701,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
       => ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_RealDef_Oreal ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(531,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0__left) ).

thf(2898,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Omonoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(1230,axiom,
    class_Rings_Osemiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring) ).

thf(4972,plain,
    class_Rings_Osemiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1230]) ).

thf(69,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zle__refl) ).

thf(1384,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(730,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_odd__less__0) ).

thf(3605,plain,
    ! [A: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(53,axiom,
    class_Groups_Oone @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oone) ).

thf(1357,plain,
    class_Groups_Oone @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(435,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
       => ? [D: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ D )
            & ! [E: $i] :
                ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ E ) ) @ D )
               => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ E )
                  & ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ E @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_lemma__interval) ).

thf(2583,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
       => ? [D: $i] :
            ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ D )
            & ! [E: $i] :
                ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ B @ E ) ) @ D )
               => ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C @ E )
                  & ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ E @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(866,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__left_Ononneg__bounded) ).

thf(4066,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ C )
          & ! [D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ B @ D ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(912,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__square) ).

thf(4239,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(416,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__less__imp__less__base) ).

thf(2507,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ B ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(1057,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_COMBB @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_c__COMBB__1) ).

thf(4659,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_COMBB @ F @ E @ D @ C @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(715,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__is__0__eq) ).

thf(3538,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
       => ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(143,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__le__trans) ).

thf(1597,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(96,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__le__mono2) ).

thf(1431,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C @ B )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(851,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult__less__le__imp__less) ).

thf(4019,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( c_Orderings_Oord__class_Oless @ E @ D @ C )
       => ( ( c_Orderings_Oord__class_Oless__eq @ E @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ D )
           => ( ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) @ B )
             => ( c_Orderings_Oord__class_Oless @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(21,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_sgn__zero) ).

thf(1299,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ A )
     => ( ( c_Groups_Osgn__class_Osgn @ A @ ( c_Groups_Ozero__class_Ozero @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(821,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__less__imp__less__left) ).

thf(3918,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
       => ( c_Orderings_Oord__class_Oless @ D @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(151,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ominus @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ominus) ).

thf(1621,plain,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ominus @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(301,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( ( c_Transcendental_Oln @ B )
            = ( c_Transcendental_Oln @ A ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__inj__iff) ).

thf(2138,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( ( ( c_Transcendental_Oln @ B )
              = ( c_Transcendental_Oln @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( c_Transcendental_Oln @ B )
              = ( c_Transcendental_Oln @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(1061,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__Re__le__cmod) ).

thf(4674,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(1148,axiom,
    class_Rings_Olinordered__semidom @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semidom) ).

thf(4826,plain,
    class_Rings_Olinordered__semidom @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).

thf(58,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_Suc__diff__diff) ).

thf(1367,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) @ B ) @ ( c_Nat_OSuc @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(1018,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__unique) ).

thf(4584,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).

thf(524,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ B @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__less__irrefl) ).

thf(2881,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ~ ( c_Orderings_Oord__class_Oless @ B @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(1004,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ B )
     => ( ( ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ B ) )
      <=> ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_of__real__eq__0__iff) ).

thf(4547,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__algebra__1 @ B )
     => ( ( ( ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) )
         => ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
        & ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
         => ( ( hAPP @ ( c_RealVector_Oof__real @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(218,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__diff__ineq) ).

thf(1848,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ C @ B ) @ ( c_RealVector_Onorm__class_Onorm @ C @ A ) ) @ ( c_RealVector_Onorm__class_Onorm @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(88,axiom,
    class_Groups_Olinordered__ab__group__add @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Olinordered__ab__group__add) ).

thf(1416,plain,
    class_Groups_Olinordered__ab__group__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(688,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__left__cancel) ).

thf(3432,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(827,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__add__one) ).

thf(3933,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
     => ( ( c_RComplete_Onatfloor @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(1001,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A )
      <=> ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_minus__less__iff) ).

thf(4533,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ B ) )
        & ( ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) @ B )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(816,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( c_RealVector_Onorm__class_Onorm @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__minus__cancel) ).

thf(3901,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) )
        = ( c_RealVector_Onorm__class_Onorm @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(564,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__0) ).

thf(3013,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(713,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_mult_Ominus__left) ).

thf(3526,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A )
        = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(834,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__1__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict) ).

thf(3960,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__1__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(1055,axiom,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ A ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__Re__le__cmod) ).

thf(4655,plain,
    ! [A: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_ORe @ A ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).

thf(461,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ A @ ( c_Groups_Ozero__class_Ozero @ A ) @ ( c_Groups_Oone__class_Oone @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zero__le__one) ).

thf(2672,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( c_Orderings_Oord__class_Oless__eq @ A @ ( c_Groups_Ozero__class_Ozero @ A ) @ ( c_Groups_Oone__class_Oone @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(523,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_zmult__zless__mono2) ).

thf(2879,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(693,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_nat__mult__assoc) ).

thf(3461,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(30,axiom,
    class_Rings_Osemiring @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring) ).

thf(1320,plain,
    class_Rings_Osemiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(252,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__neg__nonpos) ).

thf(1945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(1089,axiom,
    ~ ! [A: $i] :
        ( ( c_SEQ_Osubseq @ A )
       => ~ ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ v_f____ @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact__096_B_Bthesis_O_A_I_B_Bg_O_A_091_124_Asubseq_Ag_059_Amonoseq_A_I_Fn_O_AIm_A_Is_A_If_A_Ig_An_J_J_J_J_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096) ).

thf(4723,plain,
    ~ ! [A: $i] :
        ( ( c_SEQ_Osubseq @ A )
       => ~ ( c_SEQ_Omonoseq @ tc_RealDef_Oreal @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ tc_Nat_Onat @ c_Complex_OIm @ ( c_COMBB @ tc_Nat_Onat @ tc_Complex_Ocomplex @ tc_Nat_Onat @ v_s @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat @ v_f____ @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).

thf(1006,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_compl__le__compl__iff) ).

thf(4555,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(184,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) ).

thf(1737,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(589,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ E )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_norm__mult__less) ).

thf(3096,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ E )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) @ C )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) @ A )
         => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(325,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
      <=> ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__iff__diff__less__0) ).

thf(2206,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
         => ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
        & ( ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( c_Orderings_Oord__class_Oless @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(171,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( c_Fun_Ocomp @ G @ F @ E @ D @ C )
        = B )
     => ( ( hAPP @ D @ ( hAPP @ C @ A ) )
        = ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_o__eq__dest__lhs) ).

thf(1681,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( c_Fun_Ocomp @ G @ F @ E @ D @ C )
        = B )
     => ( ( hAPP @ D @ ( hAPP @ C @ A ) )
        = ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(526,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__diff) ).

thf(2887,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B @ A )
     => ( ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) )
        = ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(1219,axiom,
    class_Rings_Ocomm__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring) ).

thf(4949,plain,
    class_Rings_Ocomm__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1219]) ).

thf(1115,axiom,
    class_Groups_Omonoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Omonoid__add) ).

thf(4767,plain,
    class_Groups_Omonoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).

thf(284,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
    <=> ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_less__eq__real__def) ).

thf(2074,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A )
       => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
          | ( B = A ) ) )
      & ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ A )
          | ( B = A ) )
       => ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(114,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( c_Transcendental_Oln @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ B ) ) @ ( c_Transcendental_Oln @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ln__mult) ).

thf(1490,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ B )
     => ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) @ A )
       => ( ( c_Transcendental_Oln @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Transcendental_Oln @ B ) ) @ ( c_Transcendental_Oln @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(1098,axiom,
    ( ( c_RComplete_Onatceiling @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natceiling__one) ).

thf(4742,plain,
    ( ( c_RComplete_Onatceiling @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).

thf(352,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ D ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_abs__sum__triangle__ineq) ).

thf(2297,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ D ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) ) @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ C ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(937,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__le__iff__le) ).

thf(4313,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ C @ A @ B )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(691,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
      <=> ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_order__le__less) ).

thf(3443,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A )
         => ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
            | ( B = A ) ) )
        & ( ( ( c_Orderings_Oord__class_Oless @ C @ B @ A )
            | ( B = A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(1235,axiom,
    class_Rings_Ocomm__semiring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring) ).

thf(4977,plain,
    class_Rings_Ocomm__semiring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1235]) ).

thf(970,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ D @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_complex__mult) ).

thf(4423,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( c_Complex_Ocomplex_OComplex @ D @ C ) ) @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ ( c_Groups_Ominus__class_Ominus @ tc_RealDef_Oreal @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_RealDef_Oreal ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(738,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( c_Nat_OSuc @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ ( c_Nat_OSuc @ B ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
           => ( C = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__inject__base) ).

thf(3637,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ ( c_Nat_OSuc @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ A ) @ ( c_Nat_OSuc @ B ) ) )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ A )
           => ( C = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(1168,axiom,
    ( ( c_RComplete_Onatfloor @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_natfloor__one) ).

thf(4863,plain,
    ( ( c_RComplete_Onatfloor @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).

thf(314,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
      <=> ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__le__cancel__left) ).

thf(2170,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A ) )
        & ( ( c_Orderings_Oord__class_Oless__eq @ D @ B @ A )
         => ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(371,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B @ A )
      <=> ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ B @ E ) @ ( hAPP @ A @ E ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_le__fun__def) ).

thf(2351,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B @ A )
         => ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ B @ E ) @ ( hAPP @ A @ E ) ) )
        & ( ! [E: $i] : ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ B @ E ) @ ( hAPP @ A @ E ) )
         => ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(1167,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
    <=> ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_real__of__nat__less__iff) ).

thf(4858,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
       => ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A ) )
      & ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B @ A )
       => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).

thf(107,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_diff__minus__eq__add) ).

thf(1468,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( c_Groups_Ominus__class_Ominus @ C @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(761,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
          = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_neg__equal__iff__equal) ).

thf(3721,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ogroup__add @ C )
     => ( ( ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B )
            = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(891,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) ).

thf(4174,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ C ) @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(240,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
           => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ B ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_power__strict__mono) ).

thf(1912,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( c_Orderings_Oord__class_Oless @ D @ C @ B )
       => ( ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) @ C )
         => ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
           => ( c_Orderings_Oord__class_Oless @ D @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ D ) @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(571,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_ge__natfloor__plus__one__imp__gt) ).

thf(3035,plain,
    ! [A: $i,B: $i] :
      ( ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_RComplete_Onatfloor @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
     => ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(89,axiom,
    class_Rings_Ocomm__ring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring__1) ).

thf(1417,plain,
    class_Rings_Ocomm__ring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(747,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
       => ( C = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_add__right__imp__eq) ).

thf(3669,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocancel__semigroup__add @ D )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ B ) )
       => ( C = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(23063,plain,
    $false,
    inference(e,[status(thm)],[4188,3883,4835,2778,4899,4852,2480,2559,3927,4682,3021,1322,1665,1501,2452,4560,3962,1879,1337,2094,1411,2306,3345,4464,2341,4571,3053,3008,2131,4909,4262,3661,1767,1995,3930,3666,4729,3397,1899,2595,1850,3566,4728,1554,3172,1686,4166,2250,4043,3185,1924,1406,2395,1618,3377,1728,1269,4614,4714,4037,2309,1290,1882,3762,2644,3813,2381,5000,1808,3417,2590,1401,1422,1569,2248,4807,3912,3098,2676,3910,4711,4603,2099,1591,1692,4623,4973,4747,1437,4487,4592,4234,4633,4575,3332,3338,1275,3616,3253,1640,2042,3276,4026,3044,1735,1867,3175,4856,4440,2627,2168,1342,3121,4294,2901,3313,1354,2146,3895,2360,4762,1458,4941,3335,1746,1914,3502,1391,4079,3088,4916,1386,3898,4030,2173,2516,4760,4322,2136,2405,2957,3836,4343,2348,3545,2046,2972,3634,2763,2548,1782,2020,3513,4679,3308,1935,4803,3730,1645,1623,1482,4524,3938,4563,2039,4415,1907,4654,3485,4766,2745,4243,3248,1409,3263,3221,4866,4896,1298,1760,3131,3116,2667,2339,3046,3689,4224,1871,2708,3821,1323,3653,4778,3776,2682,1826,1313,1279,4275,2185,1577,1545,1922,2857,4617,4194,1319,4664,4270,4830,3470,4952,3951,4265,1815,3557,1379,3374,4622,1347,1572,4390,4471,4771,3231,4967,3268,4756,2829,3788,2222,3322,3887,4297,1418,4847,3621,1383,4686,4810,2017,4432,2007,2950,4696,4205,1415,2616,4396,1465,1302,2794,4903,4443,2105,1450,2090,5027,3393,1373,2996,4454,3295,2259,4001,4820,3163,3473,3983,3438,4147,1864,2122,3569,1677,2540,4871,3290,3093,3986,4718,2833,4842,3421,3712,1447,3808,4307,1351,4825,4536,2965,4211,4055,4230,4935,3505,1886,2424,2979,1394,4996,2242,1345,1433,4701,2750,1405,3226,4643,1943,3588,4379,4892,1697,3211,1521,1604,3011,4888,5017,2164,1708,3037,4700,1651,5012,3450,4216,4893,3591,2910,3301,2204,4753,3169,2646,4076,2737,2641,4910,2083,4100,2621,2790,2061,2773,4999,3735,1387,4466,3603,4434,4252,3610,4821,4021,4915,2409,1343,3972,4920,1355,1580,4989,2133,3752,3239,3371,3884,3134,4493,1824,4284,2150,1370,2890,4836,4160,2528,3664,3778,2015,3286,1560,1938,4849,4608,2433,3090,4417,3799,4316,4063,3282,1423,3948,3271,3166,1837,1602,2663,2332,2826,4947,3065,3750,2872,5021,1888,1296,4553,2068,1819,3144,4746,4402,1787,4942,4979,1656,2211,1407,2344,2492,3727,1281,4461,4732,2228,3647,1291,3033,1740,1402,1270,4974,4248,1375,1425,2509,4095,2638,2245,1350,3968,4808,3699,2300,3784,3831,2157,3482,1762,1534,1794,2685,2577,3381,4048,4544,4429,4636,1498,1719,4227,2537,3650,4012,3237,3941,2268,3205,4219,3018,4898,4676,4653,2023,5031,1361,4582,3596,4191,2551,4867,4457,2460,1493,4320,4951,3311,1964,3922,3407,2561,4772,4721,1461,3441,2312,2455,4750,1414,2196,2240,4744,2613,4648,1810,1791,1612,2472,4621,2649,4180,4736,1744,2781,4530,1419,2932,2845,1932,2497,1874,4259,2756,1393,4804,3242,3703,1324,2285,2519,4384,1297,3126,2836,4508,1382,2581,2465,2505,4237,1799,4717,1947,4120,1852,2813,3760,3409,4135,2766,1599,4902,1979,4689,5026,4069,3920,3364,1627,1525,3476,3755,1318,4352,4644,4625,4567,3856,1563,3999,2402,1663,1515,3508,2189,3841,4004,4995,3563,2761,3279,4889,4966,1695,4765,2883,4476,2915,4448,1648,2413,4354,4594,4874,3491,1548,4657,4393,4797,4359,3686,4489,2235,2144,4921,1278,2176,4708,4685,2292,4872,2904,3871,2033,4340,3839,3523,2734,5016,4577,2894,4365,4943,4041,4662,3070,1916,1884,3811,3519,2489,4605,4328,4381,3599,2970,3319,3234,2048,3925,2859,2938,1784,1389,2981,4309,3758,1929,4926,4677,3266,2674,1388,4801,4590,4975,3404,4612,4769,5022,1488,2959,3213,3796,4485,4558,4978,2326,3051,3330,3208,4413,1332,4060,3304,1428,3223,3118,5033,4127,2220,4796,1271,1371,4837,4645,1635,4822,1733,1997,3340,3594,2753,2442,2759,2166,1765,3619,1897,2557,4640,4887,4245,2148,2319,1356,2346,2119,4929,3815,3551,4620,3419,1542,2802,1969,3101,4790,2679,4704,3996,1408,4498,2478,3251,4369,2214,2383,2624,1894,2987,4914,4699,3732,2257,1589,2955,2299,4805,2669,2116,2839,4958,2945,4213,4616,2378,2896,3384,4773,1376,1439,2129,4569,1731,2446,1862,1325,4850,1344,2289,2252,4990,3219,2824,4203,4635,2182,5035,1471,4009,4103,1557,4731,4028,1909,4897,4985,3298,4652,4993,5007,2329,4684,2842,1952,1926,3627,2051,3806,4827,2535,1496,4890,3876,2867,4473,2770,2418,4844,4832,4720,1841,2439,1474,4232,2725,1594,3110,2086,3159,4362,3674,3953,2797,3016,1775,2592,1477,3706,3325,3142,1364,4954,2967,2390,2999,3348,2984,1805,4331,4426,4741,2952,3293,2449,3048,1340,4688,2597,4405,2816,1304,3193,3427,3936,4901,3576,2665,4886,4813,2994,5025,2318,3395,2271,3161,3672,4735,3981,2503,4177,2629,2927,3975,4812,1551,4986,4168,3391,3640,1961,3112,4673,4950,3833,3228,4971,4630,5018,2831,3988,2513,4749,3737,1770,4937,2375,4272,4939,1400,2012,3904,1941,4200,4287,4469,1780,2821,4754,1753,4894,2386,3786,1267,4437,2224,3818,4768,3655,1317,4117,3080,2747,2092,4092,4905,1637,3114,2688,3273,3202,1869,4145,2337,2276,5030,3801,3828,4132,1658,5001,2701,4241,2599,4800,4982,1679,3412,3041,3853,1336,4875,3715,2354,1358,3488,4802,4959,3415,1284,2357,2579,1504,4634,3024,2722,3366,4257,2948,3697,3124,2426,1802,3493,4624,4611,4976,1485,4106,4991,1506,4046,4932,2728,4267,2030,1372,4838,3256,2892,1272,3156,4795,4819,3684,3288,2924,4278,2458,4851,3067,1817,3874,2194,1654,2209,4646,2187,2865,2875,2435,4639,1839,4651,4138,1316,4698,4727,2088,4325,4587,4770,1966,1949,4763,2870,4944,4873,1321,1834,1517,4780,2611,4185,3056,2226,1453,2071,2658,2775,2907,4310,2792,4678,3039,1276,1326,4710,2273,4806,1341,2554,4683,4250,2811,4338,2608,1822,3613,4913,1674,2855,2853,3825,3748,2026,3516,4495,1523,4705,3859,3136,3368,4527,2416,4900,2152,4451,3085,4745,2009,4197,2912,4153,4759,3529,2162,3793,1607,3993,1661,2294,2315,2421,1689,3978,3181,2475,4855,3694,3585,3430,1777,4774,1294,2462,4387,2066,4074,4446,2739,2494,1444,4869,2917,4372,3379,2930,4039,1512,5009,1280,3907,1625,2522,3790,4367,1642,2102,4917,4114,4399,4706,3511,2155,2255,3865,2719,4254,5004,4811,2587,4596,3892,3464,1380,3521,1876,1303,4841,4579,4885,2768,1859,3328,1363,4968,3850,3479,1789,2366,1710,2714,2619,3843,4318,2201,4565,4777,4755,2334,4895,4281,4631,3804,1335,5019,4083,4024,4865,3030,3765,2063,1610,4923,4848,2847,3869,4876,4410,1902,2699,4501,2372,2159,2731,4129,1977,2430,1268,3701,2898,4972,1384,3605,1357,2583,4066,4239,2507,4659,3538,1597,1431,4019,1299,3918,1621,2138,4674,4826,1367,4584,2881,4547,1848,1416,3432,3933,4533,3901,3013,3526,3960,4655,2672,2879,3461,1320,1945,4723,4555,1737,3096,2206,1681,2887,4949,4767,2074,1490,4742,2297,4313,3443,4977,4423,3637,4863,2170,2351,4858,1468,3721,4174,1912,3035,1417,3669]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12  % Problem  : SWW205+1 : TPTP v8.2.0. Released v5.2.0.
% 0.04/0.12  % Command  : run_Leo-III %s %d THM
% 0.12/0.33  % Computer : n027.cluster.edu
% 0.12/0.33  % Model    : x86_64 x86_64
% 0.12/0.33  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33  % Memory   : 8042.1875MB
% 0.12/0.33  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33  % CPULimit : 300
% 0.12/0.33  % WCLimit  : 300
% 0.12/0.33  % DateTime : Wed Jun 19 09:04:54 EDT 2024
% 0.12/0.34  % CPUTime  : 
% 0.81/0.81  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.95/1.08  % [INFO] 	 Parsing done (269ms). 
% 1.95/1.09  % [INFO] 	 Running in sequential loop mode. 
% 2.41/1.30  % [INFO] 	 eprover registered as external prover. 
% 2.41/1.31  % [INFO] 	 Scanning for conjecture ... 
% 3.04/1.45  % [INFO] 	 Found a conjecture (or negated_conjecture) and 1273 axioms. Running axiom selection ... 
% 3.88/1.66  % [INFO] 	 Axiom selection finished. Selected 1264 axioms (removed 9 axioms). 
% 4.99/2.03  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 5.22/2.06  % [INFO] 	 Type checking passed. 
% 5.22/2.07  % [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 ... 
% 55.21/13.12  % External prover 'e' found a proof!
% 55.21/13.12  % [INFO] 	 Killing All external provers ... 
% 55.21/13.12  % Time passed: 12590ms (effective reasoning time: 12024ms)
% 55.21/13.12  % 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)>
% 55.21/13.13  % Axioms used in derivation (1264): fact_xt1_I1_J, fact_monoseq__def, fact_mult__right_OCauchy, arity_fun__Orderings_Oorder, fact_le__imp__diff__is__add, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J, fact_power__abs, fact_Re_Oadd, fact_rcis__zero__arg, fact_zle__refl, fact_minus__mult__left, fact_neg__le__0__iff__le, fact_add__le__less__mono, fact_mult__strict__left__mono__neg, fact_order__le__neq__trans, fact_Nat_Oadd__0__right, fact_real__squared__diff__one__factored, arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra, arity_Polynomial__Opoly__Rings_Osemiring__0, fact_abs__of__pos, fact_real__le__refl, fact_abs__le__D2, arity_Complex__Ocomplex__Rings_Oidom, fact_real__less__def, arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_mult__right__le__imp__le, arity_Nat__Onat__Groups_Ocancel__semigroup__add, fact_real__le__trans, fact_power__strict__decreasing, arity_Complex__Ocomplex__Groups_Omonoid__add, arity_Int__Oint__Rings_Ocomm__semiring, fact_natfloor__real__of__nat, fact_pos__add__strict, arity_Int__Oint__Rings_Osemiring, arity_RealDef__Oreal__Orderings_Oord, fact_add__strict__right__mono, arity_Int__Oint__Orderings_Olinorder, fact_incr__mult__lemma, fact_natfloor__power, fact_complex__diff__def, fact_natfloor__add, fact_mult__1__left, fact_diff__0__eq__0, fact_Im_Oconvergent, fact_nat0__intermed__int__val, fact_Complex__mult__complex__of__real, fact_Bseq__iff3, fact_diff__add__inverse, fact_of__real_Opos__bounded, fact_abs__minus__commute, fact_abs__if, fact_le__eq__less__or__eq, fact_xt1_I4_J, arity_RealDef__Oreal__Groups_Ozero, fact_trans__less__add1, fact_mult__nonneg__nonpos, fact_q__pos__lemma, fact_abs__ge__self, arity_RealDef__Oreal__Rings_Oordered__comm__semiring, arity_Complex__Ocomplex__Groups_Oab__group__add, fact_diff__0__right, fact_mult__less__cancel__left__disj, fact_nat__add__left__commute, fact_real__mult__left__cancel, fact_Bseq__monoseq__convergent, fact_mult__nonpos__nonpos, fact_order__le__less__trans, fact_abs__Re__le__cmod, fact_neg__equal__0__iff__equal, arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors, fact_power__power__power, fact_real__of__nat__zero, fact_mult__left_Ominus, fact_power__inject__base, fact_norm__power, arity_Nat__Onat__Rings_Ono__zero__divisors, fact_nat__less__cases, fact_one__is__add, fact_diff__le__mono2, arity_Int__Oint__Rings_Olinordered__semiring__strict, arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add, fact_minus__mult__right, fact_Complex__eq__1, fact_real__of__nat__diff, fact_termination__basic__simps_I5_J, arity_Polynomial__Opoly__Rings_Ocomm__semiring__0, fact_norm__mult__less, fact_odd__nonzero, fact_mult_Opos__bounded, fact_norm__diff__triangle__ineq, fact_Im_Opos__bounded, fact_zminus__zminus, fact_Re_Ononneg__bounded, fact_less__imp__le__nat, fact_le__imp__0__less, fact_add__leD1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J, fact_add__less__mono, fact_lemma__interval, fact_Bseq__minus__iff, fact_mult__eq__0__iff, fact_of__real_Ononneg__bounded, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J, arity_RealDef__Oreal__Groups_Ominus, arity_Nat__Onat__Orderings_Oorder, fact_mult__right__less__imp__less, fact_mult__left_OCauchy, fact_combine__common__factor, fact_i__mult__Complex, fact_complex__zero__def, fact_abs__eq__mult, fact_split__mult__neg__le, fact_n__less__m__mult__n, fact_ln__eq__zero__iff, fact_ln__inj__iff, fact_Suc__mono, fact_zero__less__power, arity_Nat__Onat__Rings_Ocomm__semiring__0, fact_psize__eq__0__iff, arity_Nat__Onat__Groups_Oordered__comm__monoid__add, fact_less__nat__zero__code, arity_RealDef__Oreal__Rings_Osemiring__0, arity_RealDef__Oreal__Rings_Olinordered__ring, fact_linorder__not__le, arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le, fact_mult__le__less__imp__less, fact_zadd__commute, fact_BseqD, fact_natceiling__le, fact_ln__less__cancel__iff, arity_Nat__Onat__Groups_Omonoid__mult, fact_diff__Suc__less, fact_complex__add, fact_power__increasing, fact_real__add__le__0__iff, fact_mult__0, fact_real__0__less__add__iff, fact_convergent__subseq__convergent, fact_abs__triangle__ineq2__sym, fact_order__less__asym_H, arity_Polynomial__Opoly__Rings_Olinordered__ring__strict, fact_norm__add__less, arity_Nat__Onat__Rings_Ozero__neq__one, fact_linorder__antisym__conv2, fact_zero__less__diff, fact_mult__mono, fact_complex__Re__le__cmod, fact_abs__diff__triangle__ineq, fact_leI, fact_field__power__not__zero, fact_natceiling__le__eq__one, arity_RealDef__Oreal__Rings_Oidom, fact_zero__le__natfloor, fact_order__eq__refl, fact_eq__neg__iff__add__eq__0, arity_Polynomial__Opoly__Groups_Oab__semigroup__mult, fact_zero__le__zpower__abs, fact_add__right__imp__eq, arity_Complex__Ocomplex__Groups_Oab__semigroup__mult, arity_Polynomial__Opoly__Rings_Oidom, fact_BseqI, fact_diff__diff__cancel, fact_mult_Ononneg__bounded, fact_nat__add__right__cancel, fact_neg__le__iff__le, fact_gr0__conv__Suc, arity_Polynomial__Opoly__Groups_Ominus, fact_linorder__neqE, fact_o__eq__elim, arity_RealDef__Oreal__Groups_Omonoid__add, fact_neg__0__less__iff__less, fact_mult__le__mono2, fact_Suc__diff__diff, fact_le__neq__implies__less, arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add, arity_Polynomial__Opoly__Rings_Ocomm__ring, fact_lemma__NBseq__def2, fact_zero__le__one, arity_RealDef__Oreal__Rings_Ocomm__ring__1, fact_le__minus__iff, fact_Im_OCauchy, fact_o__apply, fact_complex__Im__i, help_c__COMBC__1, fact_mult_Oadd__right, fact_mult_Ominus__right, fact_le__add__iff2, fact_mult__eq__self__implies__10, fact_complex_Osize_I2_J, fact_mult__left__le__imp__le, fact_minus__add__cancel, fact_add__pos__nonneg, fact_complex__Re__mult, fact_fun__Compl__def, fact_zle__linear, fact_complex__Im__one, arity_Int__Oint__Orderings_Oord, arity_RealDef__Oreal__Groups_Ocancel__comm__monoid__add, fact_nat__add__left__cancel__less, fact_complex__i__mult__minus, fact_int__val__lemma, fact_int__0__less__1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J, fact_linorder__antisym__conv3, fact_add__strict__increasing2, fact_order__less__imp__not__eq, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J, fact_linorder__neqE__linordered__idom, fact_mult__left_Ononneg__bounded, fact_abs__of__neg, fact_seq__suble, arity_Polynomial__Opoly__Rings_Olinordered__semidom, fact_abs__triangle__ineq2, fact_neg__less__0__iff__less, fact_mult__strict__left__mono, fact_zero__le__mult__iff, fact_natceiling__mono, fact_add__nonneg__eq__0__iff, fact_real__mult__commute, fact_real__of__nat__zero__iff, arity_Complex__Ocomplex__Rings_Ocomm__semiring__1, arity_Polynomial__Opoly__Orderings_Olinorder, fact_Re_Opos__bounded, fact_zadd__left__mono, fact_sum__squares__le__zero__iff, arity_Int__Oint__Rings_Olinordered__semiring__1__strict, fact_mult__right_Ozero, fact_mult__le__cancel__left__neg, fact_minus__real__def, fact_add__le__imp__le__left, fact_sum__squares__ge__zero, fact_add__increasing2, fact_linorder__antisym__conv1, arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add, arity_Int__Oint__Groups_Ozero, arity_RealDef__Oreal__Groups_Oabs__if, fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J, fact_mult_Ozero__left, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J, fact_i__def, arity_Int__Oint__Rings_Olinordered__ring, fact_add__0__iff, fact_le__funD, arity_Polynomial__Opoly__Orderings_Oord, fact_abs__add__one__gt__zero, fact_nat__less__real__le, fact_real__of__nat__less__iff, fact_one__less__power, fact_power__strict__increasing, fact_real__mult__assoc, fact_one__le__power, fact_even__less__0__iff, fact_self__quotient__aux1, fact_sgn__neg, fact_complex__i__not__zero, fact_power__commutes, fact_power__real__of__nat, fact_le__natfloor__eq__one, arity_Int__Oint__Groups_Ocomm__monoid__add, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J, arity_RealDef__Oreal__Groups_Omonoid__mult, fact_add__0__left, fact_add__mono, fact_complex__of__real__power, fact_gr0I, fact_sgn__minus, fact_add__right__mono, fact_i__complex__of__real, fact_le__fun__def, fact_minus__add, arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add, fact_norm__triangle__ineq3, fact_power__minus, arity_Int__Oint__Groups_Ocancel__comm__monoid__add, fact_Complex__mult__i, arity_Nat__Onat__Rings_Osemiring__0, fact_rabs__ratiotest__lemma, fact_diff__le__self, fact_lim__uminus, arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict, fact_mult__eq__1__iff, arity_Nat__Onat__Rings_Oordered__comm__semiring, fact_diff__Suc__eq__diff__pred, fact_nat__mult__1__right, arity_Complex__Ocomplex__Rings_Osemiring__0, fact_compl__mono, fact_zero__le__power__abs, fact_ord__less__eq__trans, fact_zero__reorient, 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fact_abs__add__one__not__less__self, fact_norm__minus__cancel, fact_le__add1, fact_mult__cancel1, fact_trans__le__add2, fact_abs__le__zero__iff, fact_mult__right__le__one__le, fact_mult__right_Opos__bounded, arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra, fact_sgn__1__neg, fact_of__real__eq__iff, arity_RealDef__Oreal__Groups_Oone, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J, arity_Int__Oint__Rings_Olinordered__semiring, fact_linorder__not__less, fact_Re_Ominus, fact_less__fun__def, fact_mult__diff__mult, arity_RealDef__Oreal__Groups_Ouminus, fact_realpow__num__eq__if, fact_add__less__imp__less__left, fact_sgn__pos, fact_ab__diff__minus, fact_K__record__comp, fact_Suc__mult__le__cancel1, fact_ln__less__zero, fact_abs__zero, fact_less__add__Suc1, fact_ab__left__minus, fact_sgn0, fact_add__neg__neg, fact_o__def, arity_Polynomial__Opoly__Rings_Olinordered__semiring__strict, fact_minus__diff__eq, fact_comm__semiring__class_Odistrib, fact_square__eq__iff, fact_zero__less__abs__iff, fact_abs__add__abs, fact_add__is__0, fact_nat__diff__add__eq1, fact_termination__basic__simps_I3_J, fact_Im_Ononneg__bounded, arity_Int__Oint__Rings_Oordered__ring, fact_abs__le__iff, fact_incseq__imp__monoseq, fact_power_Opower_Opower__Suc, fact_complex__diff, fact_abs__triangle__ineq4, fact_o__eq__dest__lhs, fact_sgn__zero__iff, fact_Deriv_Oadd__diff__add, fact_mult__right__mono, fact_xt1_I2_J, fact_abs__minus__cancel, arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add, fact_add__0, fact_Cauchy__convergent__iff, fact_diff__minus__eq__add, fact_power__mono, fact_lemmaCauchy, fact_mult__le__mono1, fact_add__eq__0__iff, fact_neg__equal__iff__equal, fact_compl__eq__compl__iff, arity_Polynomial__Opoly__Rings_Olinordered__semiring__1, arity_RealDef__Oreal__Groups_Ocomm__monoid__mult, fact_diff__add__0, fact_add__pos__pos, arity_Nat__Onat__Groups_Oab__semigroup__mult, fact_diff__cancel2, arity_Polynomial__Opoly__Rings_Oordered__ring, fact_nat__less__add__iff2, fact_add__nonneg__nonneg, fact_nat__eq__add__iff1, arity_RealDef__Oreal__Power_Opower, arity_RealDef__Oreal__Groups_Oab__group__add, fact_real__of__nat__power, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J, fact_fun__diff__def, fact_diff__self__eq__0, arity_Int__Oint__Rings_Olinordered__idom, arity_Int__Oint__Rings_Ocomm__semiring__1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J, fact_self__quotient__aux2, arity_RealDef__Oreal__Rings_Olinordered__semidom, fact_sgn__greater, fact_realpow__two__disj, arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1, fact_mult__nonpos__nonneg, fact_Suc__diff__le, fact_Bseq__iff2, arity_Polynomial__Opoly__Rings_Osemiring, fact_order__eq__iff, arity_RealDef__Oreal__Rings_Olinordered__semiring__1__strict, arity_Nat__Onat__Groups_Oab__semigroup__add, fact_of__real__diff, fact_nat__mult__eq__1__iff, fact_power__mult__distrib, arity_Int__Oint__Rings_Oordered__comm__semiring, fact_zmult__zless__mono2, fact_Suc__pred, fact_ge__natfloor__plus__one__imp__gt, arity_Int__Oint__Groups_Oab__semigroup__add, arity_fun__Groups_Ominus, arity_Int__Oint__Groups_Oone, help_c__COMBK__1, arity_Nat__Onat__Rings_Olinordered__semidom, fact_add__le__cancel__right, arity_Complex__Ocomplex__Groups_Omonoid__mult, arity_Int__Oint__Groups_Ocomm__monoid__mult, fact_add__left__mono, fact_real__add__eq__0__iff, fact_n__less__n__mult__m, fact_mult__right_Ominus, fact_less__not__refl, fact_double__add__le__zero__iff__single__add__le__zero, fact_mult__less__cancel__left__neg, fact_zabs__def, fact_ord__eq__less__trans, fact_sgn__if, fact_less__1__mult, fact_compl__le__compl__iff, fact_le0, fact_power__one, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J, arity_Int__Oint__Orderings_Oorder, fact_Cauchy__iff, fact_le__diff__conv2, fact_mult__pos__pos, fact_sgn__real__def, fact_of__real__mult, fact_minus__mult__minus, fact_ex__least__nat__less, fact_crossproduct__eq, fact_decseq__imp__monoseq, fact_square__eq__1__iff, fact_le__trans, arity_Int__Oint__Groups_Oabs__if, fact_diff__Suc__diff__eq1, fact_comm__mult__left__mono, fact_gr__implies__not0, fact_mult__strict__right__mono__neg, fact_decr__lemma, fact_Im_Odiff, fact_abs__triangle__ineq, fact_zsgn__def, arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add, fact_le__mult__natfloor, arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors, fact_mult__less__imp__less__right, fact_zero__neq__one, fact_power__eq__imp__eq__base, fact_norm__mult__ineq, fact_diff__eq__diff__less__eq, fact_real__Cauchy__convergent, fact_Re_OisUCont, fact_natceiling__subtract, fact_Complex__eq__0, fact_diff__Suc__1, fact_add__less__le__mono, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J, arity_Polynomial__Opoly__Orderings_Opreorder, fact_real__of__nat__Suc__gt__zero, fact_real__two__squares__add__zero__iff, fact_power__less__power__Suc, fact_of__real_Ominus, fact_less__add__one, fact_int__one__le__iff__zero__less, fact_Cauchy__subseq__Cauchy, arity_Nat__Onat__Rings_Olinordered__semiring__strict, fact_natfloor__add__one, fact_nat__add__commute, arity_Int__Oint__Rings_Ono__zero__divisors, fact_mult__neg__neg, arity_RealDef__Oreal__Rings_Ocomm__semiring__0, fact_Complex__add__complex__of__real, fact_add__leE, arity_Int__Oint__Rings_Oring__1, fact_right__minus__eq, arity_RealDef__Oreal__Groups_Ocancel__ab__semigroup__add, fact_neg__0__le__iff__le, fact_mult_Odiff__left, fact_ln__realpow, fact_zmult__zminus, fact_zdiv__mono2__lemma, arity_Polynomial__Opoly__Groups_Ogroup__add, fact_sgn__one, arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add, fact_Im, arity_RealDef__Oreal__Rings_Oring__1, fact_isUCont__Cauchy, fact_nat__mult__commute, fact_zero__less__Suc, fact_add1__zle__eq, fact_Cauchy__Bseq, fact_ln__less__zero__iff, arity_Complex__Ocomplex__Rings_Ocomm__ring, fact_mult_Ocomm__neutral, fact_real__add__left__mono, fact_add__strict__increasing, fact_real__mult__order, arity_RealDef__Oreal__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, arity_RealDef__Oreal__Rings_Oordered__semiring, arity_Polynomial__Opoly__Rings_Olinordered__semiring, fact_eq__imp__le, fact_diff__add__inverse2, arity_Int__Oint__Rings_Olinordered__semiring__1, fact_plus__nat_Oadd__0, fact_subseq__Suc__iff, fact_minus__le__self__iff, fact_diff__eq__diff__eq, fact_order__le__imp__less__or__eq, fact_add__nonpos__neg, fact_mult__left__idem, fact_complex__of__real__i, fact_real__norm__def, fact_of__real_Ozero, fact_minus__equation__iff, fact_Im_Ozero, fact_le__antisym, fact_zadd__0__right, arity_Int__Oint__Rings_Olinordered__comm__semiring__strict, fact_of__real__power, fact_diff__le__mono, arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring, fact_of__real_Oconvergent, arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le, arity_Polynomial__Opoly__Rings_Ocomm__ring__1, fact_expi__zero, fact_order__antisym__conv, fact_neq0__conv, fact_mult__left_Opos__bounded, fact_mult__left__mono__neg, fact_real__of__nat__gt__zero__cancel__iff, fact_mult__0__right, fact_real__add__less__0__iff, fact_zless__linear, fact_mult__less__cancel__left__pos, fact_real__mult__le__cancel__iff2, fact_trans__le__add1, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J, fact_monoseq__realpow, fact_abs__mult__pos, fact_complex__one__def, fact_zero__less__mult__pos, arity_Polynomial__Opoly__Rings_Ocomm__semiring, arity_Int__Oint__Groups_Ominus, arity_Complex__Ocomplex__Rings_Omult__zero, fact_zmult__1__right, arity_Polynomial__Opoly__Rings_Ono__zero__divisors, fact_double__eq__0__iff, fact_le__0__eq, fact_mult__Suc__right, fact_realpow__pos__nth__unique, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J, fact_mult__less__cancel2, fact_zle__add1__eq__le, fact_mult__1__right, fact_zero__less__two, fact_diff__is__0__eq, fact_decseq__eq__incseq, fact_add__le__imp__le__right, fact_order__less__irrefl, arity_Nat__Onat__Rings_Osemiring, arity_Polynomial__Opoly__Groups_Oabs__if, fact_order__less__imp__le, fact_termination__basic__simps_I1_J, fact_real__0__le__add__iff, fact_order__less__trans, fact_real__of__nat__one, arity_Polynomial__Opoly__Groups_Ozero, fact_rp, fact_real__of__nat__le__iff, fact_abs__mult__self, fact_real__mult__less__mono2, fact_sum__squares__eq__zero__iff, fact_Re_Oconvergent, fact_abs__norm__cancel, fact_le__add__diff__inverse, fact_nat__zero__less__power__iff, fact_zabs__less__one__iff, arity_Int__Oint__Rings_Ocomm__ring__1, fact_add__le__mono, fact_less__Suc0, fact_convergent__minus__iff, fact_complex__of__real__add__Complex, fact_order__le__less, fact_add__diff__inverse, fact_of__real__0, fact_mult__right_Oconvergent, 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_mult__less__imp__less__left, fact_le__diff__iff, arity_Nat__Onat__Groups_Oone, fact_diff__Suc__Suc, fact_add__imp__eq, arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J, fact_power__gt1, fact_nat__less__le, fact_le__square, fact_zadd__zmult__distrib, fact_divisors__zero, arity_Polynomial__Opoly__Groups_Oone, arity_RealDef__Oreal__Groups_Olinordered__ab__group__add, fact_ab__semigroup__add__class_Oadd__ac_I1_J, arity_RealDef__Oreal__Groups_Oab__semigroup__mult, arity_Int__Oint__Rings_Ozero__neq__one, fact_ln__gt__zero, arity_RealDef__Oreal__Rings_Oring__1__no__zero__divisors, arity_Complex__Ocomplex__RealVector_Oreal__normed__vector, fact_no__zero__divisors, fact_sgn__1__pos, fact_less__iff__diff__less__0, fact_mult__eq__if, fact_Bseq__ignore__initial__segment, fact_le__cube, fact_zless__add1__eq, fact_linorder__cases, fact_real__natfloor__le, fact_xt1_I11_J, fact_add__less__mono1, fact_add__le__cancel__left, arity_Polynomial__Opoly__Rings_Olinordered__ring, fact_mult__sgn__abs, arity_RealDef__Oreal__Groups_Ogroup__add, arity_Polynomial__Opoly__Groups_Ocomm__monoid__add, fact_real__le__linear, arity_Complex__Ocomplex__Power_Opower, fact_ord__eq__le__trans, fact_of__real_OisUCont, fact_f_I1_J, fact_Re_Obounded, fact_zadd__zminus__inverse2, fact_o__assoc, fact_double__add__less__zero__iff__single__add__less__zero, fact_less__diff__iff, fact_real__natfloor__gt__diff__one, fact_odd__less__0, arity_Int__Oint__Rings_Ocomm__semiring__0, fact_complex__add__def, fact_power__le__imp__le__base, fact_complex_Orecs, fact_real__of__nat__inject, fact_norm__triangle__ineq2, arity_Int__Oint__Groups_Omonoid__add, fact_expi__add, fact_real__of__nat__add, fact_ln__gt__zero__iff, fact_xt1_I7_J, fact_order__trans, arity_Complex__Ocomplex__Groups_Ozero, arity_Complex__Ocomplex__Groups_Ocancel__comm__monoid__add, fact_not__square__less__zero, arity_Complex__Ocomplex__Groups_Ominus, fact_mult__1, fact_complex__of__real__def, fact_Bseq__def, fact_less__bin__lemma, arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add, arity_Int__Oint__Rings_Olinordered__ring__strict, fact_mult_Odiff__right, fact_zdiv__mono2__neg__lemma, fact_ln__one, fact_real__sgn__def, fact_natfloor__mono, fact_ln__ge__zero__iff, arity_Complex__Ocomplex__Rings_Ocomm__semiring__0, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J, fact_mult__left_Oconvergent, arity_Int__Oint__Orderings_Opreorder
% 55.21/13.13  % No. of inferences in proof: 2532
% 55.21/13.13  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 12590 ms resp. 12024 ms w/o parsing
% 56.58/13.55  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 56.58/13.57  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------