TSTP Solution File: SWW206+1 by Leo-III---1.7.15

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.15
% Problem  : SWW206+1 : TPTP v8.2.0. Released v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d THM

% Computer : n025.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 18:21:19 EDT 2024

% Result   : Theorem 57.34s 14.28s
% Output   : Refutation 58.39s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      : 1266
% Syntax   : Number of formulae    : 2534 ( 830 unt;   0 typ;   0 def)
%            Number of atoms       : 6319 (1620 equ;   0 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 41028 ( 292   ~; 127   |; 488   &;36951   @)
%                                         ( 232 <=>;2938  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (   8 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  126 ( 124 usr;  13 con; 0-5 aty)
%            Number of variables   : 5434 (   0   ^5339   !;  95   ?;5434   :)

% 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_OIm_type,type,
    c_Complex_OIm: $i ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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__strict_type,type,
    class_Rings_Olinordered__semiring__1__strict: $i > $o ).

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

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

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

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

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

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

thf(class_Groups_Ominus_type,type,
    class_Groups_Ominus: $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(c_SEQ_Osubseq_type,type,
    c_SEQ_Osubseq: $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(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_Rings_Ozero__neq__one_type,type,
    class_Rings_Ozero__neq__one: $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_Rings_Oring__1_type,type,
    class_Rings_Oring__1: $i > $o ).

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

thf(class_Rings_Olinordered__ring__strict_type,type,
    class_Rings_Olinordered__ring__strict: $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_Groups_Ocancel__ab__semigroup__add_type,type,
    class_Groups_Ocancel__ab__semigroup__add: $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_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_Groups_Olinordered__ab__group__add_type,type,
    class_Groups_Olinordered__ab__group__add: $i > $o ).

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

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

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

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

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

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

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

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_SEQ_Odecseq_type,type,
    c_SEQ_Odecseq: $i > $i > $o ).

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

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

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

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

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

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

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

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

thf(2630,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[446]) ).

thf(839,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_split__mult__pos__le) ).

thf(3962,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[839]) ).

thf(51,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zless__imp__add1__zle) ).

thf(1411,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[51]) ).

thf(442,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__less__cases) ).

thf(2612,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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)],[442]) ).

thf(1147,axiom,
    class_Groups_Oordered__ab__group__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__group__add) ).

thf(4835,plain,
    class_Groups_Oordered__ab__group__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).

thf(485,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__triangle__sub) ).

thf(2778,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[485]) ).

thf(114,axiom,
    class_Groups_Osgn__if @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Osgn__if) ).

thf(1559,plain,
    class_Groups_Osgn__if @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(860,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__minus__le__complex__mod) ).

thf(4022,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[860]) ).

thf(686,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__le__mono1) ).

thf(3434,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[686]) ).

thf(386,axiom,
    ( ( hAPP @ c_Complex_ORe @ c_Complex_Oii )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__i) ).

thf(2412,plain,
    ( ( hAPP @ c_Complex_ORe @ c_Complex_Oii )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(723,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).

thf(3566,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)],[723]) ).

thf(1174,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) ).

thf(4877,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).

thf(312,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Power_Opower @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Power_Opower) ).

thf(2199,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Power_Opower @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(560,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ ( c_RComplete_Onatfloor @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__natfloor) ).

thf(3021,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ ( c_RComplete_Onatfloor @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(87,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ B ) ) )
      <=> ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__less__Suc__eq) ).

thf(1501,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ B ) ) )
         => ( A = B ) )
        & ( ( A = B )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(918,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
      <=> ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
            & ? [D: $i] :
              ! [E: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__iff2) ).

thf(4201,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OBseq @ B @ A )
         => ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ? [D: $i] :
                ! [E: $i] : ( hBOOL @ ( hAPP @ ( 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] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ? [D: $i] :
                ! [E: $i] : ( hBOOL @ ( hAPP @ ( 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)],[918]) ).

thf(396,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/sandbox2/benchmark/theBenchmark.p',fact_real__mult__1) ).

thf(2452,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)],[396]) ).

thf(853,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/sandbox2/benchmark/theBenchmark.p',fact_abs__minus__add__cancel) ).

thf(4005,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)],[853]) ).

thf(1018,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
            & ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ C ) ) )
      <=> ? [C: $i] :
          ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_lemma__NBseq__def2) ).

thf(4571,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ C ) ) )
         => ? [C: $i] :
            ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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] : ( hBOOL @ ( hAPP @ ( 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] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[1018]) ).

thf(570,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__strict__increasing) ).

thf(3053,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[570]) ).

thf(358,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Oring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring) ).

thf(2336,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Oring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(294,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/sandbox2/benchmark/theBenchmark.p',fact_neg__equal__zero) ).

thf(2131,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)],[294]) ).

thf(1195,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__mult) ).

thf(4909,plain,
    class_Groups_Oab__semigroup__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).

thf(289,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__add__less1) ).

thf(2109,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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)],[289]) ).

thf(1139,axiom,
    class_Groups_Ouminus @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ouminus) ).

thf(4824,plain,
    class_Groups_Ouminus @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).

thf(677,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos2) ).

thf(3399,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[677]) ).

thf(338,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ A @ ( c_Groups_Oone__class_Oone @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__one__le__zero) ).

thf(2263,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( hBOOL @ ( hAPP @ ( 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)],[338]) ).

thf(912,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/sandbox2/benchmark/theBenchmark.p',fact_sgn0) ).

thf(4183,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)],[912]) ).

thf(943,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/sandbox2/benchmark/theBenchmark.p',fact_mult__cancel2) ).

thf(4290,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)],[943]) ).

thf(844,axiom,
    ! [A: $i] :
      ( ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        | ( hBOOL @ ( hAPP @ ( 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 ) ) )
        | ( hBOOL @ ( hAPP @ ( 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 ) ) )
        | ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_unimodular__reduce__norm) ).

thf(3979,plain,
    ! [A: $i] :
      ( ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A )
        = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        | ( hBOOL @ ( hAPP @ ( 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 ) ) )
        | ( hBOOL @ ( hAPP @ ( 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 ) ) )
        | ( hBOOL @ ( hAPP @ ( 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)],[844]) ).

thf(203,axiom,
    ( ( hAPP @ c_Complex_Ocnj @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj__one) ).

thf(1840,plain,
    ( ( hAPP @ c_Complex_Ocnj @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(1159,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__0) ).

thf(4852,plain,
    class_Rings_Ocomm__semiring__0 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).

thf(489,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__group__add) ).

thf(2787,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)],[489]) ).

thf(436,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/sandbox2/benchmark/theBenchmark.p',fact_power__mult__distrib) ).

thf(2595,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)],[436]) ).

thf(407,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    <=> ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( B
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__power__eq__Suc__0__iff) ).

thf(2499,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( A
            = ( 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_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(424,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__is__0__eq_H) ).

thf(2559,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[424]) ).

thf(857,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_trans__le__add1) ).

thf(4015,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[857]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__less__imp__less__base) ).

thf(1950,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(1246,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/sandbox2/benchmark/theBenchmark.p',fact_complex__add__def) ).

thf(4992,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)],[1246]) ).

thf(111,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__trans__Suc) ).

thf(1554,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(1190,axiom,
    c_SEQ_Osubseq @ v_f____,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_f_I1_J) ).

thf(4904,plain,
    c_SEQ_Osubseq @ v_f____,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).

thf(906,axiom,
    ? [A: $i] :
    ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Im_Obounded) ).

thf(4166,plain,
    ? [A: $i] :
    ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[906]) ).

thf(333,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Groups_Oab__semigroup__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__mult) ).

thf(2250,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)],[333]) ).

thf(848,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/sandbox2/benchmark/theBenchmark.p',fact_o__eq__elim) ).

thf(3990,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)],[848]) ).

thf(317,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__le) ).

thf(2210,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(927,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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Odiff) ).

thf(4226,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)],[927]) ).

thf(443,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/sandbox2/benchmark/theBenchmark.p',fact_abs__idempotent) ).

thf(2622,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)],[443]) ).

thf(622,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/sandbox2/benchmark/theBenchmark.p',fact_diff__add__inverse2) ).

thf(3230,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)],[622]) ).

thf(717,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__add__eq2) ).

thf(3549,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[717]) ).

thf(1252,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/sandbox2/benchmark/theBenchmark.p',fact_complex_Osize_I1_J) ).

thf(5000,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)],[1252]) ).

thf(638,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) ).

thf(3285,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)],[638]) ).

thf(601,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__add__one__not__less__self) ).

thf(3153,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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)],[601]) ).

thf(824,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl) ).

thf(3912,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(170,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/sandbox2/benchmark/theBenchmark.p',fact_right__minus) ).

thf(1728,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)],[170]) ).

thf(1071,axiom,
    class_Groups_Oordered__comm__monoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add) ).

thf(4711,plain,
    class_Groups_Oordered__comm__monoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(3,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ ( c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mono) ).

thf(1269,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ ( c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(969,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semidom) ).

thf(4408,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(1092,axiom,
    class_Groups_Olinordered__ab__group__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Olinordered__ab__group__add) ).

thf(4747,plain,
    class_Groups_Olinordered__ab__group__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).

thf(370,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I1_J) ).

thf(2363,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(209,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/sandbox2/benchmark/theBenchmark.p',fact_power__minus) ).

thf(1855,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)],[209]) ).

thf(1213,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Im__cnj) ).

thf(4931,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1213]) ).

thf(867,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__mult__pos) ).

thf(4043,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[867]) ).

thf(713,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).

thf(3534,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)],[713]) ).

thf(1074,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocomm__monoid__mult) ).

thf(4714,plain,
    class_Groups_Ocomm__monoid__mult @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(349,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
            & ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ C ) ) )
      <=> ? [C: $i] :
          ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ 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/sandbox2/benchmark/theBenchmark.p',fact_lemma__NBseq__def) ).

thf(2309,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ C ) ) )
         => ? [C: $i] :
            ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ 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] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_Nat_OSuc @ C ) ) ) )
         => ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[349]) ).

thf(409,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__le__add__iff2) ).

thf(2512,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[409]) ).

thf(50,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add1__zle__eq) ).

thf(1406,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[50]) ).

thf(287,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__cancel) ).

thf(2104,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[287]) ).

thf(877,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__nonpos__neg) ).

thf(4075,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[877]) ).

thf(217,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hAPP @ B @ C )
          = ( hAPP @ A @ C ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ext) ).

thf(1882,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hAPP @ B @ C )
          = ( hAPP @ A @ C ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(497,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_LIMSEQ__inverse__realpow__zero__lemma) ).

thf(2808,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[497]) ).

thf(95,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osimps_I2_J) ).

thf(1522,plain,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(453,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_q__neg__lemma) ).

thf(2644,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[453]) ).

thf(119,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__self) ).

thf(1569,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(332,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I7_J) ).

thf(2248,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(1127,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__comm__monoid__add) ).

thf(4807,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).

thf(1060,axiom,
    class_Rings_Ocomm__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring) ).

thf(4697,plain,
    class_Rings_Ocomm__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(838,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) ).

thf(3959,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)],[838]) ).

thf(461,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/sandbox2/benchmark/theBenchmark.p',fact_minus__minus) ).

thf(2671,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)],[461]) ).

thf(463,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__le__cancel__iff1) ).

thf(2676,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[463]) ).

thf(823,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/sandbox2/benchmark/theBenchmark.p',fact_i__def) ).

thf(3910,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)],[823]) ).

thf(704,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__double__add__iff__zero__le__single__add) ).

thf(3498,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[704]) ).

thf(376,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq3) ).

thf(2380,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[376]) ).

thf(249,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__of__nonneg) ).

thf(1971,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[249]) ).

thf(100,axiom,
    class_Rings_Oring__no__zero__divisors @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__no__zero__divisors) ).

thf(1533,plain,
    class_Rings_Oring__no__zero__divisors @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(78,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sgn__greater) ).

thf(1486,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[78]) ).

thf(1008,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__nat_Odiff__0) ).

thf(4532,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)],[1008]) ).

thf(154,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq) ).

thf(1671,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[154]) ).

thf(684,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj__power) ).

thf(3429,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(465,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/sandbox2/benchmark/theBenchmark.p',fact_minus__equation__iff) ).

thf(2686,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)],[465]) ).

thf(21,axiom,
    class_Rings_Olinordered__semiring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1) ).

thf(1310,plain,
    class_Rings_Olinordered__semiring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(834,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_eq__diff__iff) ).

thf(3942,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[834]) ).

thf(689,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__le__trans) ).

thf(3446,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(734,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/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__assoc) ).

thf(3616,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)],[734]) ).

thf(1016,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Omonoseq @ B @ A )
      <=> ( ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ ( c_Nat_OSuc @ C ) ) ) )
          | ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ ( c_Nat_OSuc @ C ) ) ) @ ( hAPP @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_monoseq__Suc) ).

thf(4564,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( ( c_SEQ_Omonoseq @ B @ A )
         => ( ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ ( c_Nat_OSuc @ C ) ) ) )
            | ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ ( c_Nat_OSuc @ C ) ) ) @ ( hAPP @ A @ C ) ) ) ) )
        & ( ( ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ ( c_Nat_OSuc @ C ) ) ) )
            | ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ ( c_Nat_OSuc @ C ) ) ) @ ( hAPP @ A @ C ) ) ) )
         => ( c_SEQ_Omonoseq @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(527,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ? [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( 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 ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reals__Archimedean6) ).

thf(2901,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ? [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( 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 ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(1236,axiom,
    class_Rings_Osemiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring) ).

thf(4980,plain,
    class_Rings_Osemiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1236]) ).

thf(1099,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/sandbox2/benchmark/theBenchmark.p',fact_lim__uminus) ).

thf(4762,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)],[1099]) ).

thf(636,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/sandbox2/benchmark/theBenchmark.p',fact_power__0__left) ).

thf(3276,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)],[636]) ).

thf(84,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__leI) ).

thf(1497,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(310,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__trans) ).

thf(2195,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(60,axiom,
    class_Orderings_Oord @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Oord) ).

thf(1426,plain,
    class_Orderings_Oord @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(620,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__eq__less) ).

thf(3217,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(788,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/sandbox2/benchmark/theBenchmark.p',fact_right__minus__eq) ).

thf(3794,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)],[788]) ).

thf(1037,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ B ) ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj__mult) ).

thf(4633,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ B ) ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(688,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__left__disj) ).

thf(3439,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) )
        & ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[688]) ).

thf(1151,axiom,
    class_Groups_Omonoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__add) ).

thf(4839,plain,
    class_Groups_Omonoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).

thf(655,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).

thf(3332,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)],[655]) ).

thf(666,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__mono_H) ).

thf(3367,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[666]) ).

thf(1179,axiom,
    ( ( c_RComplete_Onatfloor @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__one) ).

thf(4884,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)],[1179]) ).

thf(1170,axiom,
    class_RealVector_Oreal__normed__vector @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__vector) ).

thf(4870,plain,
    class_RealVector_Oreal__normed__vector @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).

thf(531,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add) ).

thf(2918,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)],[531]) ).

thf(213,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/sandbox2/benchmark/theBenchmark.p',fact_o__eq__dest__lhs) ).

thf(1867,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)],[213]) ).

thf(306,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Odecseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_decseq__imp__monoseq) ).

thf(2184,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)],[306]) ).

thf(1039,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( B != A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__neq__trans) ).

thf(4638,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( B != A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(214,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/sandbox2/benchmark/theBenchmark.p',fact_fun__Compl__def) ).

thf(1872,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)],[214]) ).

thf(783,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).

thf(3781,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)],[783]) ).

thf(978,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 ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ E ) @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__lemma) ).

thf(4440,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 ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ E ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(1043,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left__idem) ).

thf(4650,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)],[1043]) ).

thf(400,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Ozero) ).

thf(2476,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)],[400]) ).

thf(445,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/sandbox2/benchmark/theBenchmark.p',fact_real__squared__diff__one__factored) ).

thf(2627,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)],[445]) ).

thf(301,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I2_J) ).

thf(2168,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[301]) ).

thf(872,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Ocomm__monoid__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult) ).

thf(4058,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)],[872]) ).

thf(231,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__ge__zero) ).

thf(1919,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[231]) ).

thf(629,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__lessD1) ).

thf(3253,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(255,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/sandbox2/benchmark/theBenchmark.p',fact_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(1988,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)],[255]) ).

thf(31,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__comm__monoid__add) ).

thf(1342,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(592,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__less__power__Suc) ).

thf(3121,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[592]) ).

thf(1255,axiom,
    class_Rings_Ocomm__ring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__ring) ).

thf(5005,plain,
    class_Rings_Ocomm__ring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1255]) ).

thf(144,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Ozero__left) ).

thf(1640,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)],[144]) ).

thf(88,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/sandbox2/benchmark/theBenchmark.p',fact_double__compl) ).

thf(1507,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)],[88]) ).

thf(816,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__nat_Osimps_I1_J) ).

thf(3895,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[816]) ).

thf(768,axiom,
    ! [A: $i] :
      ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__cnj) ).

thf(3717,plain,
    ! [A: $i] :
      ( ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(1217,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/sandbox2/benchmark/theBenchmark.p',fact_complex__add) ).

thf(4941,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)],[1217]) ).

thf(656,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono) ).

thf(3335,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[656]) ).

thf(1119,axiom,
    class_Rings_Olinordered__ring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__ring) ).

thf(4792,plain,
    class_Rings_Olinordered__ring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).

thf(661,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/sandbox2/benchmark/theBenchmark.p',fact_nat__add__right__cancel) ).

thf(3350,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)],[661]) ).

thf(545,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__mult__less) ).

thf(2972,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[545]) ).

thf(1052,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__le__less__linear) ).

thf(4679,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(1031,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__square) ).

thf(4618,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[1031]) ).

thf(134,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__Suc) ).

thf(1613,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[134]) ).

thf(232,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__ring) ).

thf(1921,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(863,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__1__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict) ).

thf(4030,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)],[863]) ).

thf(384,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__less__le__imp__less) ).

thf(2405,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[384]) ).

thf(515,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_even__less__0__iff) ).

thf(2869,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[515]) ).

thf(542,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/sandbox2/benchmark/theBenchmark.p',fact_times_Oidem) ).

thf(2957,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)],[542]) ).

thf(175,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/sandbox2/benchmark/theBenchmark.p',fact_left__add__mult__distrib) ).

thf(1746,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)],[175]) ).

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

thf(3103,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(798,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Ononneg__bounded) ).

thf(3836,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[798]) ).

thf(320,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__less__mono2) ).

thf(2216,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[320]) ).

thf(624,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oabs__if @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
            = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__if) ).

thf(3236,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oabs__if @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[624]) ).

thf(391,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natceiling__neg) ).

thf(2432,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[391]) ).

thf(849,axiom,
    ( ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal )
   != ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__zero__not__eq__one) ).

thf(3994,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)],[849]) ).

thf(825,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/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__1__right) ).

thf(3915,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)],[825]) ).

thf(421,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__cancel__left) ).

thf(2548,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[421]) ).

thf(996,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/sandbox2/benchmark/theBenchmark.p',fact_plus__nat_Oadd__0) ).

thf(4496,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)],[996]) ).

thf(324,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) ).

thf(2227,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)],[324]) ).

thf(627,axiom,
    ? [A: $i] :
    ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Re_Obounded) ).

thf(3249,plain,
    ? [A: $i] :
    ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[627]) ).

thf(937,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( hAPP @ c_Complex_ORe @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ ( hAPP @ c_Complex_OIm @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj__def) ).

thf(4266,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ A )
      = ( c_Complex_Ocomplex_OComplex @ ( 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)],[937]) ).

thf(647,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 ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D ) @ F ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__neg__lemma) ).

thf(3308,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 ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D ) @ F ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(238,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/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__eq__zero__iff) ).

thf(1935,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)],[238]) ).

thf(128,axiom,
    class_Groups_Oordered__ab__group__add__abs @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__group__add__abs) ).

thf(1601,plain,
    class_Groups_Oordered__ab__group__add__abs @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(694,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).

thf(3466,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)],[694]) ).

thf(917,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__left__pos) ).

thf(4198,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[917]) ).

thf(279,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/sandbox2/benchmark/theBenchmark.p',fact_nat__add__assoc) ).

thf(2082,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)],[279]) ).

thf(382,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/sandbox2/benchmark/theBenchmark.p',fact_mult__Suc) ).

thf(2400,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)],[382]) ).

thf(1097,axiom,
    class_Rings_Oring__no__zero__divisors @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__no__zero__divisors) ).

thf(4760,plain,
    class_Rings_Oring__no__zero__divisors @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).

thf(1227,axiom,
    class_Rings_Ocomm__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring) ).

thf(4963,plain,
    class_Rings_Ocomm__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1227]) ).

thf(247,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I10_J) ).

thf(1967,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(951,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4322,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)],[951]) ).

thf(500,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__ge__minus__self) ).

thf(2819,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( 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)],[500]) ).

thf(190,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/sandbox2/benchmark/theBenchmark.p',fact_divisors__zero) ).

thf(1803,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)],[190]) ).

thf(225,axiom,
    ! [A: $i] :
      ( ( c_RComplete_Onatceiling @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__real__of__nat) ).

thf(1904,plain,
    ! [A: $i] :
      ( ( c_RComplete_Onatceiling @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(93,axiom,
    class_Groups_Oone @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oone) ).

thf(1518,plain,
    class_Groups_Oone @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(162,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/sandbox2/benchmark/theBenchmark.p',fact_eq__add__iff2) ).

thf(1703,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)],[162]) ).

thf(1266,axiom,
    class_Rings_Osemiring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring) ).

thf(5024,plain,
    class_Rings_Osemiring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1266]) ).

thf(219,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq4) ).

thf(1887,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[219]) ).

thf(930,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__times) ).

thf(4238,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)],[930]) ).

thf(11,axiom,
    class_Groups_Ouminus @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ouminus) ).

thf(1287,plain,
    class_Groups_Ouminus @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(163,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/sandbox2/benchmark/theBenchmark.p',fact_abs__norm__cancel) ).

thf(1709,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)],[163]) ).

thf(590,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/sandbox2/benchmark/theBenchmark.p',fact_add__0) ).

thf(3116,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)],[590]) ).

thf(341,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__Suc__ex__iff) ).

thf(2269,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(703,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__add__iff1) ).

thf(3495,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[703]) ).

thf(1098,axiom,
    class_Rings_Olinordered__semiring__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__strict) ).

thf(4761,plain,
    class_Rings_Olinordered__semiring__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).

thf(226,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/sandbox2/benchmark/theBenchmark.p',fact_abs__add__abs) ).

thf(1907,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)],[226]) ).

thf(27,axiom,
    class_Rings_Olinordered__ring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring) ).

thf(1330,plain,
    class_Rings_Olinordered__ring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(90,axiom,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__n__not__n) ).

thf(1511,plain,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(1109,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/sandbox2/benchmark/theBenchmark.p',fact_complex__zero__def) ).

thf(4778,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)],[1109]) ).

thf(122,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__diff1__eq) ).

thf(1577,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[122]) ).

thf(790,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__zero__less__power__iff) ).

thf(3803,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[790]) ).

thf(367,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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__one) ).

thf(2356,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)],[367]) ).

thf(843,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/sandbox2/benchmark/theBenchmark.p',fact_uminus__apply) ).

thf(3976,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)],[843]) ).

thf(530,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__neg__neg) ).

thf(2916,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[530]) ).

thf(1005,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less0) ).

thf(4524,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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)],[1005]) ).

thf(612,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) ) @ ( c_RComplete_Onatceiling @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__mono) ).

thf(3195,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[612]) ).

thf(562,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ A ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__natfloor__le) ).

thf(3029,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[562]) ).

thf(1089,axiom,
    class_RealVector_Oreal__normed__algebra__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__algebra__1) ).

thf(4743,plain,
    class_RealVector_Oreal__normed__algebra__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).

thf(188,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__less__cancel1) ).

thf(1792,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[188]) ).

thf(743,axiom,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__abs__def) ).

thf(3644,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[743]) ).

thf(16,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__assoc) ).

thf(1298,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)],[16]) ).

thf(178,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__imp__less__right) ).

thf(1760,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(1223,axiom,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(4956,plain,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1223]) ).

thf(716,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__is__0__eq) ).

thf(3542,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( 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)],[716]) ).

thf(24,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__left__commute) ).

thf(1313,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)],[24]) ).

thf(957,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).

thf(4349,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)],[957]) ).

thf(425,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__less__iff1) ).

thf(2562,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[425]) ).

thf(404,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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sgn__1__neg) ).

thf(2488,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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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)],[404]) ).

thf(1017,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__less__nonneg) ).

thf(4568,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(1240,axiom,
    class_RealVector_Oreal__normed__algebra__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1) ).

thf(4984,plain,
    class_RealVector_Oreal__normed__algebra__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1240]) ).

thf(1101,axiom,
    class_RealVector_Oreal__normed__div__algebra @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra) ).

thf(4766,plain,
    class_RealVector_Oreal__normed__div__algebra @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).

thf(71,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/sandbox2/benchmark/theBenchmark.p',fact_compl__eq__compl__iff) ).

thf(1462,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)],[71]) ).

thf(709,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/sandbox2/benchmark/theBenchmark.p',fact_mult__zero__right) ).

thf(3517,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)],[709]) ).

thf(383,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__le__D2) ).

thf(2403,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(1048,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_metric__bound__lemma) ).

thf(4664,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[1048]) ).

thf(73,axiom,
    class_Rings_Oordered__ring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring) ).

thf(1479,plain,
    class_Rings_Oordered__ring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(266,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__add__strict) ).

thf(2024,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[266]) ).

thf(549,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_th) ).

thf(2989,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ ( hAPP @ c_Complex_OIm @ ( 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)],[549]) ).

thf(609,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/sandbox2/benchmark/theBenchmark.p',fact_nat__1__eq__mult__iff) ).

thf(3180,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)],[609]) ).

thf(1105,axiom,
    class_Groups_Oordered__comm__monoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__comm__monoid__add) ).

thf(4771,plain,
    class_Groups_Oordered__comm__monoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).

thf(504,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__neg__neg) ).

thf(2829,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[504]) ).

thf(512,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/sandbox2/benchmark/theBenchmark.p',fact_natfloor__power) ).

thf(2857,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)],[512]) ).

thf(1165,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/sandbox2/benchmark/theBenchmark.p',fact_complex__one__def) ).

thf(4862,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)],[1165]) ).

thf(1210,axiom,
    class_Rings_Ono__zero__divisors @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ono__zero__divisors) ).

thf(4928,plain,
    class_Rings_Ono__zero__divisors @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1210]) ).

thf(33,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__add) ).

thf(1347,plain,
    class_Groups_Ocomm__monoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(121,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ ( c_Nat_OSuc @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__less__eq) ).

thf(1572,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ ( c_Nat_OSuc @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ ( c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(1144,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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__minus__iff) ).

thf(4830,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)],[1144]) ).

thf(949,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/sandbox2/benchmark/theBenchmark.p',fact_i__mult__eq2) ).

thf(4317,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)],[949]) ).

thf(773,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/sandbox2/benchmark/theBenchmark.p',fact_complex__i__mult__minus) ).

thf(3734,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)],[773]) ).

thf(1221,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/sandbox2/benchmark/theBenchmark.p',fact_isUCont__Cauchy) ).

thf(4952,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)],[1221]) ).

thf(984,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).

thf(4460,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)],[984]) ).

thf(83,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/sandbox2/benchmark/theBenchmark.p',help_c__COMBS__1) ).

thf(1494,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)],[83]) ).

thf(1264,axiom,
    ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Im__one) ).

thf(5020,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)],[1264]) ).

thf(720,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__add__assoc2) ).

thf(3557,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[720]) ).

thf(215,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 ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__less) ).

thf(1875,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(1120,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/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__rcis) ).

thf(4793,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)],[1120]) ).

thf(1000,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/sandbox2/benchmark/theBenchmark.p',fact_double__zero__sym) ).

thf(4507,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)],[1000]) ).

thf(945,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__antisym) ).

thf(4302,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(379,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im) ).

thf(2388,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(416,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C ) @ E ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ D ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__quotient__lemma) ).

thf(2533,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C ) @ E ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ D ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(933,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/sandbox2/benchmark/theBenchmark.p',fact_minus__add__cancel) ).

thf(4253,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)],[933]) ).

thf(1231,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_RealDef_Oreal @ tc_Nat_Onat @ B )
        = ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__inject) ).

thf(4967,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)],[1231]) ).

thf(854,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq2) ).

thf(4008,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[854]) ).

thf(631,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/sandbox2/benchmark/theBenchmark.p',fact_convergent__add) ).

thf(3258,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)],[631]) ).

thf(305,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__less__norm__iff) ).

thf(2178,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[305]) ).

thf(651,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__natfloor) ).

thf(3322,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(240,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oord @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Oord) ).

thf(1946,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oord @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(711,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
      <=> ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__0__iff) ).

thf(3524,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) ) ) )
        & ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[711]) ).

thf(492,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq4) ).

thf(2794,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[492]) ).

thf(750,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/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib2) ).

thf(3670,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)],[750]) ).

thf(762,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__trans) ).

thf(3702,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(1134,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ A ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__In__mult__cnj__zero) ).

thf(4815,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_OIm @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Complex_Ocomplex ) @ A ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).

thf(263,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__neg__pos) ).

thf(2017,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[263]) ).

thf(79,axiom,
    class_Rings_Ozero__neq__one @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ozero__neq__one) ).

thf(1489,plain,
    class_Rings_Ozero__neq__one @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(1059,axiom,
    class_Rings_Oordered__cancel__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__cancel__semiring) ).

thf(4696,plain,
    class_Rings_Oordered__cancel__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(328,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__le__add__iff1) ).

thf(2237,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[328]) ).

thf(1243,axiom,
    ! [A: $i] :
      ( ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_Complex_Ocomplex @ tc_Nat_Onat @ c_Complex_Ocnj @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Oconvergent) ).

thf(4988,plain,
    ! [A: $i] :
      ( ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_Oconvergent @ tc_Complex_Ocomplex @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_Complex_Ocomplex @ tc_Nat_Onat @ c_Complex_Ocnj @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1243]) ).

thf(54,axiom,
    class_Rings_Olinordered__idom @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__idom) ).

thf(1415,plain,
    class_Rings_Olinordered__idom @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(813,axiom,
    ( ( hAPP @ c_Complex_Ocnj @ c_Complex_Oii )
    = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj__i) ).

thf(3887,plain,
    ( ( hAPP @ c_Complex_Ocnj @ c_Complex_Oii )
    = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ c_Complex_Oii ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(1021,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Ozero__right) ).

thf(4583,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)],[1021]) ).

thf(165,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_less__1__mult) ).

thf(1714,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[165]) ).

thf(909,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__pos__pos) ).

thf(4173,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[909]) ).

thf(831,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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__mult) ).

thf(3934,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)],[831]) ).

thf(1189,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__0) ).

thf(4903,plain,
    class_Rings_Ocomm__semiring__0 @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).

thf(979,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/sandbox2/benchmark/theBenchmark.p',fact_rcis__mult) ).

thf(4443,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)],[979]) ).

thf(876,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__add__assoc) ).

thf(4072,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[876]) ).

thf(67,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_ln__inj__iff) ).

thf(1450,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[67]) ).

thf(282,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ B ) ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Oadd) ).

thf(2090,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ B ) ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(675,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/sandbox2/benchmark/theBenchmark.p',fact_complex__equality) ).

thf(3393,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)],[675]) ).

thf(1162,axiom,
    class_Rings_Osemiring__0 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Osemiring__0) ).

thf(4857,plain,
    class_Rings_Osemiring__0 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).

thf(935,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__le__cases) ).

thf(4258,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(41,axiom,
    class_Groups_Omonoid__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__mult) ).

thf(1373,plain,
    class_Groups_Omonoid__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(983,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv1) ).

thf(4454,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(1027,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 ) )
         => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
            & ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_sgn__if) ).

thf(4600,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 ) )
         => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
            & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[1027]) ).

thf(642,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_not__square__less__zero) ).

thf(3295,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ~ ( hBOOL @ ( hAPP @ ( 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)],[642]) ).

thf(487,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__add__left__mono) ).

thf(2782,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[487]) ).

thf(1137,axiom,
    class_Rings_Osemiring__0 @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Osemiring__0) ).

thf(4820,plain,
    class_Rings_Osemiring__0 @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).

thf(604,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ozero__neq__one @ A )
     => ( ( c_Groups_Ozero__class_Ozero @ A )
       != ( c_Groups_Oone__class_Oone @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__neq__one) ).

thf(3163,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)],[604]) ).

thf(56,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__eq) ).

thf(1418,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ B ) ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(692,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/sandbox2/benchmark/theBenchmark.p',fact_add__eq__self__zero) ).

thf(3453,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)],[692]) ).

thf(519,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( c_RComplete_Onatfloor @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__natfloor__eq__one) ).

thf(2880,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( c_RComplete_Onatfloor @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[519]) ).

thf(157,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv2) ).

thf(1682,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(210,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 ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__less__eq) ).

thf(1858,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(899,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__add__less__0__iff) ).

thf(4147,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[899]) ).

thf(534,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).

thf(2925,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)],[534]) ).

thf(842,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/sandbox2/benchmark/theBenchmark.p',fact_norm__minus__cancel) ).

thf(3973,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)],[842]) ).

thf(896,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__imp__less__left) ).

thf(4134,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(662,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__less__imp__less) ).

thf(3357,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(335,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__minus__iff) ).

thf(2254,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(724,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__1) ).

thf(3569,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)],[724]) ).

thf(1171,axiom,
    ( ( c_RComplete_Onatfloor @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__zero) ).

thf(4871,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)],[1171]) ).

thf(644,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/sandbox2/benchmark/theBenchmark.p',fact_abs__mult) ).

thf(3300,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)],[644]) ).

thf(584,axiom,
    ( ( c_Transcendental_Oln @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__one) ).

thf(3093,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)],[584]) ).

thf(423,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/sandbox2/benchmark/theBenchmark.p',fact_o__eq__dest) ).

thf(2555,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)],[423]) ).

thf(640,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/sandbox2/benchmark/theBenchmark.p',fact_add_Ocomm__neutral) ).

thf(3290,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)],[640]) ).

thf(8,axiom,
    class_Groups_Oordered__ab__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__semigroup__add) ).

thf(1283,plain,
    class_Groups_Oordered__ab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(277,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/sandbox2/benchmark/theBenchmark.p',fact_add__right__cancel) ).

thf(2073,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)],[277]) ).

thf(781,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/sandbox2/benchmark/theBenchmark.p',fact_zero__reorient) ).

thf(3771,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)],[781]) ).

thf(1129,axiom,
    class_Groups_Oab__semigroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__add) ).

thf(4810,plain,
    class_Groups_Oab__semigroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).

thf(904,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Ocomm__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring) ).

thf(4162,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)],[904]) ).

thf(804,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__add__le__0__iff) ).

thf(3851,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[804]) ).

thf(1078,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
       => ( c_SEQ_OBseq @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Cauchy__Bseq) ).

thf(4718,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)],[1078]) ).

thf(871,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/sandbox2/benchmark/theBenchmark.p',fact_add__imp__eq) ).

thf(4055,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)],[871]) ).

thf(1154,axiom,
    class_Groups_Ogroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ogroup__add) ).

thf(4842,plain,
    class_Groups_Ogroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).

thf(223,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__le__eq__one) ).

thf(1896,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[223]) ).

thf(766,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__ge__zero__imp__ge__one) ).

thf(3712,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[766]) ).

thf(1248,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4996,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1248]) ).

thf(1140,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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__zero) ).

thf(4825,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)],[1140]) ).

thf(544,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__iff__gr__or__eq) ).

thf(2965,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) )
            | ( B = A ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) )
            | ( B = A ) )
         => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(1186,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__add) ).

thf(4892,plain,
    class_Groups_Ocomm__monoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).

thf(921,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/sandbox2/benchmark/theBenchmark.p',fact_real__diff__def) ).

thf(4211,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)],[921]) ).

thf(990,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/sandbox2/benchmark/theBenchmark.p',fact_norm__mult) ).

thf(4475,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)],[990]) ).

thf(371,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).

thf(2367,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)],[371]) ).

thf(714,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Olinordered__ab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add) ).

thf(3537,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)],[714]) ).

thf(235,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Rings_Ocomm__semiring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__1) ).

thf(1928,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)],[235]) ).

thf(610,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr0I) ).

thf(3189,plain,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[610]) ).

thf(344,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/sandbox2/benchmark/theBenchmark.p',fact_real__mult__left__cancel) ).

thf(2281,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)],[344]) ).

thf(999,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Opos__bounded) ).

thf(4504,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[999]) ).

thf(707,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/sandbox2/benchmark/theBenchmark.p',fact_one__reorient) ).

thf(3505,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)],[707]) ).

thf(755,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__pos__neg) ).

thf(3685,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[755]) ).

thf(565,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring) ).

thf(3036,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(415,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/sandbox2/benchmark/theBenchmark.p',fact_diff__def) ).

thf(2530,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)],[415]) ).

thf(96,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ominus @ B )
     => ( class_Groups_Ominus @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Groups_Ominus) ).

thf(1526,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ominus @ B )
     => ( class_Groups_Ominus @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(947,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Complex_Ocnj @ B )
        = ( hAPP @ c_Complex_Ocnj @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj__cancel__iff) ).

thf(4307,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ c_Complex_Ocnj @ B )
          = ( hAPP @ c_Complex_Ocnj @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ c_Complex_Ocnj @ B )
          = ( hAPP @ c_Complex_Ocnj @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(752,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Ominus__left) ).

thf(3676,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)],[752]) ).

thf(295,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/sandbox2/benchmark/theBenchmark.p',fact_zpower__zpower) ).

thf(2137,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)],[295]) ).

thf(866,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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__add) ).

thf(4040,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)],[866]) ).

thf(664,axiom,
    ( c_Complex_Oii
   != ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__i__not__zero) ).

thf(3361,plain,
    ( c_Complex_Oii
   != ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(259,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).

thf(2002,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)],[259]) ).

thf(626,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Omonoseq @ B @ A )
      <=> ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ D ) ) ) )
          | ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) ) @ ( hAPP @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_monoseq__def) ).

thf(3243,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( ( c_SEQ_Omonoseq @ B @ A )
         => ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
               => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ D ) ) ) )
            | ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
               => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) ) @ ( hAPP @ A @ C ) ) ) ) ) )
        & ( ( ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
               => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ D ) ) ) )
            | ! [C: $i,D: $i] :
                ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
               => ( hBOOL @ ( hAPP @ ( 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)],[626]) ).

thf(494,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/sandbox2/benchmark/theBenchmark.p',fact_mult__1__right) ).

thf(2799,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)],[494]) ).

thf(865,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_not__sum__squares__lt__zero) ).

thf(4038,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ~ ( hBOOL @ ( hAPP @ ( 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)],[865]) ).

thf(63,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__le) ).

thf(1433,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
          & ( B != A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
          & ( B != A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(1064,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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__ignore__initial__segment) ).

thf(4701,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)],[1064]) ).

thf(1041,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natfloor__neg) ).

thf(4643,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[1041]) ).

thf(239,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/sandbox2/benchmark/theBenchmark.p',fact_left__minus) ).

thf(1943,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)],[239]) ).

thf(1069,axiom,
    class_Groups_Osgn__if @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Osgn__if) ).

thf(4707,plain,
    class_Groups_Osgn__if @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(807,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ D ) )
            & ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ E ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lemma__interval) ).

thf(3870,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ D ) )
            & ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ E ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(298,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel1) ).

thf(2154,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[298]) ).

thf(821,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__pos__nonneg) ).

thf(3906,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[821]) ).

thf(702,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/sandbox2/benchmark/theBenchmark.p',fact_minus__real__def) ).

thf(3492,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)],[702]) ).

thf(680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__antisym__conv) ).

thf(3406,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
           => ( A = B ) )
          & ( ( A = B )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(880,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/sandbox2/benchmark/theBenchmark.p',fact_add__left__imp__eq) ).

thf(4082,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)],[880]) ).

thf(993,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__add__iff2) ).

thf(4483,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[993]) ).

thf(211,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__comm__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__comm__semiring) ).

thf(1863,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)],[211]) ).

thf(261,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ D ) )
            & ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ E ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lemma__interval__lt) ).

thf(2011,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ D ) )
            & ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ C ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ E ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(1160,axiom,
    class_Rings_Olinordered__semiring__strict @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring__strict) ).

thf(4853,plain,
    class_Rings_Olinordered__semiring__strict @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).

thf(556,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I6_J) ).

thf(3011,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(645,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less) ).

thf(3303,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[645]) ).

thf(159,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__nonneg__nonneg) ).

thf(1691,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[159]) ).

thf(712,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/sandbox2/benchmark/theBenchmark.p',fact_Deriv_Oadd__diff__add) ).

thf(3531,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)],[712]) ).

thf(292,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__less__iff) ).

thf(2118,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(29,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Nat__Transfer_Otsub @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) ).

thf(1338,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Nat__Transfer_Otsub @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(476,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__strict__mono) ).

thf(2737,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[476]) ).

thf(1196,axiom,
    hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact__096cmod_A_Is_A0_J_A_060_061_Ar_096) ).

thf(4910,plain,
    hBOOL @ ( hAPP @ ( 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)],[1196]) ).

thf(406,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__le__0__iff__le) ).

thf(2496,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[406]) ).

thf(753,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_not__real__of__nat__less__zero) ).

thf(3679,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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)],[753]) ).

thf(1063,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring__1) ).

thf(4700,plain,
    class_Rings_Ocomm__semiring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(148,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/sandbox2/benchmark/theBenchmark.p',fact_diff__minus__eq__add) ).

thf(1651,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)],[148]) ).

thf(818,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_incr__lemma) ).

thf(3899,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[818]) ).

thf(533,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/sandbox2/benchmark/theBenchmark.p',fact_ab__diff__minus) ).

thf(2922,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)],[533]) ).

thf(691,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__of__nonpos) ).

thf(3450,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[691]) ).

thf(706,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Ometric__space @ B )
     => ( ( c_SEQ_Oconvergent @ B @ A )
       => ( c_SEQ_OCauchy @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_convergent__Cauchy) ).

thf(3503,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)],[706]) ).

thf(923,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__less__add__iff2) ).

thf(4216,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[923]) ).

thf(1187,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/sandbox2/benchmark/theBenchmark.p',fact_Complex__eq__1) ).

thf(4893,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)],[1187]) ).

thf(1023,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__increasing) ).

thf(4589,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[1023]) ).

thf(1035,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__mult__ineq) ).

thf(4629,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[1035]) ).

thf(430,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__add__less) ).

thf(2574,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ E @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[430]) ).

thf(1182,axiom,
    class_Rings_Omult__zero @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Omult__zero) ).

thf(4888,plain,
    class_Rings_Omult__zero @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).

thf(769,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/sandbox2/benchmark/theBenchmark.p',fact_real__mult__assoc) ).

thf(3720,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)],[769]) ).

thf(607,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__le) ).

thf(3169,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
          & ( B != A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
          & ( B != A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(992,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/sandbox2/benchmark/theBenchmark.p',fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) ).

thf(4480,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)],[992]) ).

thf(454,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__less) ).

thf(2646,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
            | ( B = A ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
            | ( B = A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__ge__zero) ).

thf(1915,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[229]) ).

thf(283,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel__left__pos) ).

thf(2093,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[283]) ).

thf(22,axiom,
    class_Rings_Omult__zero @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Omult__zero) ).

thf(1311,plain,
    class_Rings_Omult__zero @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(326,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__le__eq) ).

thf(2232,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(458,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add) ).

thf(2660,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)],[458]) ).

thf(180,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__less__mono1) ).

thf(1768,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[180]) ).

thf(377,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) ) @ ( c_Transcendental_Oln @ A ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__le__cancel__iff) ).

thf(2382,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) ) @ ( c_Transcendental_Oln @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[377]) ).

thf(484,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel__less) ).

thf(2773,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[484]) ).

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

thf(4999,plain,
    class_Groups_Oab__group__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1251]) ).

thf(258,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oring__1__no__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors) ).

thf(2000,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)],[258]) ).

thf(1201,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_Complex_Ocomplex @ tc_Nat_Onat @ c_Complex_Ocnj @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_OCauchy) ).

thf(4915,plain,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ A )
     => ( c_SEQ_OCauchy @ tc_Complex_Ocomplex @ ( c_COMBB @ tc_Complex_Ocomplex @ tc_Complex_Ocomplex @ tc_Nat_Onat @ c_Complex_Ocnj @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).

thf(1072,axiom,
    class_Groups_Ozero @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ozero) ).

thf(4712,plain,
    class_Groups_Ozero @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(1205,axiom,
    class_Rings_Oidom @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oidom) ).

thf(4920,plain,
    class_Rings_Oidom @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1205]) ).

thf(965,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( ( c_Nat__Transfer_Otsub @ A @ B )
          = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ B ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( ( c_Nat__Transfer_Otsub @ A @ B )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tsub__def) ).

thf(4375,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( ( c_Nat__Transfer_Otsub @ A @ B )
          = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ B ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( ( c_Nat__Transfer_Otsub @ A @ B )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(1175,axiom,
    class_Rings_Omult__zero @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Omult__zero) ).

thf(4878,plain,
    class_Rings_Omult__zero @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).

thf(156,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) )
        <=> ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__antisym__conv3) ).

thf(1676,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) )
           => ( A = B ) )
          & ( ( A = B )
           => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(152,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__ge__zero) ).

thf(1666,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( hBOOL @ ( hAPP @ ( 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)],[152]) ).

thf(197,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( ( B != A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__cases) ).

thf(1824,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( ( B != A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(1138,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/sandbox2/benchmark/theBenchmark.p',fact_complex_Osize_I2_J) ).

thf(4821,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)],[1138]) ).

thf(115,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_odd__less__0) ).

thf(1560,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[115]) ).

thf(32,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Zero__neq__Suc) ).

thf(1343,plain,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(496,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/sandbox2/benchmark/theBenchmark.p',fact_minus__add) ).

thf(2805,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)],[496]) ).

thf(94,axiom,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint )
   != ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__0__neq__1) ).

thf(1519,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)],[94]) ).

thf(806,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/sandbox2/benchmark/theBenchmark.p',fact_norm__zero) ).

thf(3867,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)],[806]) ).

thf(776,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/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel__disj) ).

thf(3742,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)],[776]) ).

thf(956,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__le__iff) ).

thf(4344,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[956]) ).

thf(883,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__mult__pos) ).

thf(4091,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(777,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__diff__triangle__ineq) ).

thf(3752,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ E )
     => ( hBOOL @ ( hAPP @ ( 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)],[777]) ).

thf(1004,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__ratiotest__lemma) ).

thf(4521,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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)],[1004]) ).

thf(623,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__zero__iff) ).

thf(3233,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[623]) ).

thf(668,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__le__cancel1) ).

thf(3371,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[668]) ).

thf(812,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/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__left) ).

thf(3884,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)],[812]) ).

thf(1113,axiom,
    c_SEQ_Osubseq @ v_ga____,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_g_I1_J) ).

thf(4785,plain,
    c_SEQ_Osubseq @ v_ga____,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).

thf(997,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oring__no__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors) ).

thf(4499,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)],[997]) ).

thf(981,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__group__add__abs @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__group__add__abs) ).

thf(4449,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)],[981]) ).

thf(596,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/sandbox2/benchmark/theBenchmark.p',fact_ab__left__minus) ).

thf(3134,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)],[596]) ).

thf(995,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__0__right) ).

thf(4493,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)],[995]) ).

thf(522,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__le__cmod) ).

thf(2890,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[522]) ).

thf(387,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__less__add1) ).

thf(2414,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[387]) ).

thf(1148,axiom,
    class_Rings_Osemiring__0 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Osemiring__0) ).

thf(4836,plain,
    class_Rings_Osemiring__0 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).

thf(903,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__strict__increasing2) ).

thf(4160,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[903]) ).

thf(1047,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/sandbox2/benchmark/theBenchmark.p',fact_minus__unique) ).

thf(4661,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)],[1047]) ).

thf(414,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq2__sym) ).

thf(2528,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[414]) ).

thf(748,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) ).

thf(3664,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)],[748]) ).

thf(616,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff) ).

thf(3207,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[616]) ).

thf(754,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1__strict @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_convex__bound__lt) ).

thf(3682,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1__strict @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( hBOOL @ ( hAPP @ ( 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)],[754]) ).

thf(782,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/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__1) ).

thf(3778,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)],[782]) ).

thf(262,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Ocomm__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring) ).

thf(2015,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring @ A )
     => ( class_Rings_Ocomm__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(814,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
        | ( B = A )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__less__linear) ).

thf(3889,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
        | ( B = A )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(605,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__ge__zero) ).

thf(3166,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( 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)],[605]) ).

thf(129,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A )
      = ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__Suc__le__raw) ).

thf(1602,plain,
    ! [A: $i] :
      ( ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A )
      = ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(1028,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__less__add__iff1) ).

thf(4608,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[1028]) ).

thf(1107,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__semigroup__add) ).

thf(4776,plain,
    class_Groups_Ocancel__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).

thf(195,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__neg__nonpos) ).

thf(1819,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[195]) ).

thf(599,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_split__mult__neg__le) ).

thf(3144,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[599]) ).

thf(1235,axiom,
    class_Rings_Olinordered__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semiring) ).

thf(4979,plain,
    class_Rings_Olinordered__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1235]) ).

thf(355,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/sandbox2/benchmark/theBenchmark.p',fact_zmult__zminus) ).

thf(2327,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)],[355]) ).

thf(363,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__idom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oabs__class_Oabs @ D @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) ) @ C ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__diff__less__iff) ).

thf(2345,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__idom @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oabs__class_Oabs @ D @ ( c_Groups_Ominus__class_Ominus @ D @ C @ B ) ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) ) @ C ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ominus__class_Ominus @ D @ B @ A ) ) @ C ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ B ) @ A ) ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[363]) ).

thf(455,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) ) @ ( hAPP @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__funD) ).

thf(2653,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) ) @ ( hAPP @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__not__less) ).

thf(3282,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(57,axiom,
    class_Groups_Ogroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ogroup__add) ).

thf(1423,plain,
    class_Groups_Ogroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(835,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__0__less__add__iff) ).

thf(3948,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[835]) ).

thf(39,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
     => ( ( c_Nat__Transfer_Otsub @ A @ B )
        = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tsub__eq) ).

thf(1365,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
     => ( ( c_Nat__Transfer_Otsub @ A @ B )
        = ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(915,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__0__le__add__iff) ).

thf(4192,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[915]) ).

thf(1081,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/sandbox2/benchmark/theBenchmark.p',fact_Complex__eq__0) ).

thf(4725,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)],[1081]) ).

thf(583,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj__cnj) ).

thf(3090,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(426,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/sandbox2/benchmark/theBenchmark.p',fact_field__power__not__zero) ).

thf(2565,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)],[426]) ).

thf(503,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__minus) ).

thf(2826,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)],[503]) ).

thf(901,axiom,
    ( ( hAPP @ c_Complex_Ocnj @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Ozero) ).

thf(4155,plain,
    ( ( hAPP @ c_Complex_Ocnj @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(649,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add) ).

thf(3314,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)],[649]) ).

thf(643,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__add__le__zero__iff__single__add__le__zero) ).

thf(3297,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[643]) ).

thf(1095,axiom,
    class_Groups_Oone @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oone) ).

thf(4757,plain,
    class_Groups_Oone @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).

thf(1219,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/sandbox2/benchmark/theBenchmark.p',fact_complex__minus) ).

thf(4947,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)],[1219]) ).

thf(597,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__eq__le__trans) ).

thf(3137,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(575,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__strict__right__mono) ).

thf(3065,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[575]) ).

thf(516,axiom,
    ( c_Complex_Oii
   != ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__i__not__one) ).

thf(2872,plain,
    ( c_Complex_Oii
   != ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(967,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/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel) ).

thf(4385,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)],[967]) ).

thf(201,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__quotient__lemma__neg) ).

thf(1836,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(186,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Osemiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring) ).

thf(1787,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Osemiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(362,axiom,
    hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J) ).

thf(2344,plain,
    hBOOL @ ( hAPP @ ( 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)],[362]) ).

thf(779,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__diff__conv) ).

thf(3763,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[779]) ).

thf(888,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/sandbox2/benchmark/theBenchmark.p',fact_diff__0__eq__0) ).

thf(4104,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)],[888]) ).

thf(141,axiom,
    class_Groups_Olinordered__ab__group__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Olinordered__ab__group__add) ).

thf(1634,plain,
    class_Groups_Olinordered__ab__group__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(47,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__commute) ).

thf(1392,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)],[47]) ).

thf(749,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__left__neg) ).

thf(3667,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[749]) ).

thf(290,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leD2) ).

thf(2112,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(13,axiom,
    class_Rings_Ocomm__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring) ).

thf(1291,plain,
    class_Rings_Ocomm__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(879,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__le__square) ).

thf(4080,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__ring @ B )
     => ( hBOOL @ ( hAPP @ ( 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)],[879]) ).

thf(830,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/sandbox2/benchmark/theBenchmark.p',fact_diff__add__0) ).

thf(3931,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)],[830]) ).

thf(564,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/sandbox2/benchmark/theBenchmark.p',fact_nat__add__commute) ).

thf(3033,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)],[564]) ).

thf(173,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj) ).

thf(1740,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = ( c_Complex_Ocomplex_OComplex @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(227,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__increasing__iff) ).

thf(1910,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[227]) ).

thf(106,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__le__lessD) ).

thf(1541,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(291,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/sandbox2/benchmark/theBenchmark.p',fact_minus__diff__eq) ).

thf(2115,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)],[291]) ).

thf(182,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Oab__group__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__group__add) ).

thf(1772,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)],[182]) ).

thf(375,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__zmult__distrib) ).

thf(2377,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)],[375]) ).

thf(1232,axiom,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_ORe,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_OisUCont) ).

thf(4974,plain,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_ORe,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1232]) ).

thf(671,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__diff__conv2) ).

thf(3381,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[671]) ).

thf(469,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono1) ).

thf(2709,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[469]) ).

thf(1038,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__power__ineq) ).

thf(4636,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__algebra__1 @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[1038]) ).

thf(591,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono2) ).

thf(3119,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[591]) ).

thf(856,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__antisym) ).

thf(4012,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(450,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__refl) ).

thf(2638,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(924,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__pos__neg2) ).

thf(4219,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[924]) ).

thf(553,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__less__add2) ).

thf(3001,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[553]) ).

thf(1128,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/sandbox2/benchmark/theBenchmark.p',fact_Im_OCauchy) ).

thf(4808,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)],[1128]) ).

thf(893,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
          & ! [C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult_Opos__bounded) ).

thf(4123,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
          & ! [C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[893]) ).

thf(331,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Oadd__left) ).

thf(2245,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)],[331]) ).

thf(588,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/sandbox2/benchmark/theBenchmark.p',fact_add__is__0) ).

thf(3105,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)],[588]) ).

thf(347,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__left__mono) ).

thf(2300,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[347]) ).

thf(784,axiom,
    ! [A: $i,B: $i] :
      ( ( c_SEQ_Osubseq @ B )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_seq__suble) ).

thf(3784,plain,
    ! [A: $i,B: $i] :
      ( ( c_SEQ_Osubseq @ B )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(253,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/sandbox2/benchmark/theBenchmark.p',fact_power_Opower_Opower__0) ).

thf(1983,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)],[253]) ).

thf(1224,axiom,
    class_Groups_Ominus @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ominus) ).

thf(4957,plain,
    class_Groups_Ominus @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1224]) ).

thf(179,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__le__zero__iff) ).

thf(1762,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[179]) ).

thf(410,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__right__mono) ).

thf(2515,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[410]) ).

thf(101,axiom,
    class_Orderings_Olinorder @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Olinorder) ).

thf(1534,plain,
    class_Orderings_Olinorder @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(1256,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/sandbox2/benchmark/theBenchmark.p',fact_Re_Oconvergent) ).

thf(5006,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)],[1256]) ).

thf(43,axiom,
    class_Rings_Ocomm__ring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring__1) ).

thf(1375,plain,
    class_Rings_Ocomm__ring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(994,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel2) ).

thf(4486,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[994]) ).

thf(1116,axiom,
    class_Rings_Oordered__cancel__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__cancel__semiring) ).

thf(4789,plain,
    class_Rings_Oordered__cancel__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).

thf(975,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_realpow__pos__nth) ).

thf(4429,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( 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)],[975]) ).

thf(59,axiom,
    class_Rings_Osemiring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring) ).

thf(1425,plain,
    class_Rings_Osemiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(1157,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/sandbox2/benchmark/theBenchmark.p',fact_psize__eq__0__iff) ).

thf(4845,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)],[1157]) ).

thf(408,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/sandbox2/benchmark/theBenchmark.p',fact_add__mult__distrib) ).

thf(2509,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)],[408]) ).

thf(929,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
        | ( B = A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__or__eq__imp__le) ).

thf(4233,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
        | ( B = A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(763,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/sandbox2/benchmark/theBenchmark.p',fact_mult__1) ).

thf(3704,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)],[763]) ).

thf(1212,axiom,
    class_Rings_Ono__zero__divisors @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ono__zero__divisors) ).

thf(4930,plain,
    class_Rings_Ono__zero__divisors @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1212]) ).

thf(167,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__cancel__left) ).

thf(1719,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[167]) ).

thf(418,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Oone__class_Oone @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__Suc__less) ).

thf(2537,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( c_Groups_Oone__class_Oone @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[418]) ).

thf(1152,axiom,
    class_Rings_Ocomm__ring__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring__1) ).

thf(4840,plain,
    class_Rings_Ocomm__ring__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).

thf(1178,axiom,
    class_Groups_Ozero @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ozero) ).

thf(4883,plain,
    class_Groups_Ozero @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).

thf(573,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) ).

thf(3060,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[573]) ).

thf(582,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__less__0__iff__less) ).

thf(3087,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[582]) ).

thf(559,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/sandbox2/benchmark/theBenchmark.p',fact_power__one) ).

thf(3018,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)],[559]) ).

thf(118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zadd__zless__mono) ).

thf(1567,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[118]) ).

thf(399,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
    <=> ~ ( ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__split__asm) ).

thf(2460,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
       => ~ ( ( ( hBOOL @ ( hAPP @ ( 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_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)],[399]) ).

thf(1003,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/sandbox2/benchmark/theBenchmark.p',fact_complex_Orecs) ).

thf(4518,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)],[1003]) ).

thf(38,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Zero__not__Suc) ).

thf(1361,plain,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(246,axiom,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Re_Ononneg__bounded) ).

thf(1964,plain,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[246]) ).

thf(14,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__less__cancel1) ).

thf(1292,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[14]) ).

thf(1106,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/sandbox2/benchmark/theBenchmark.p',fact__096_B_Bthesis_O_A_I_B_Bg_O_A_091_124_Asubseq_Ag_059_Amonoseq_A_I_Fn_O_AIm_A_Is_A_If_A_Ig_An_J_J_J_J_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096) ).

thf(4772,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)],[1106]) ).

thf(767,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/sandbox2/benchmark/theBenchmark.p',help_c__COMBC__1) ).

thf(3714,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)],[767]) ).

thf(1001,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/sandbox2/benchmark/theBenchmark.p',fact_complex__mult) ).

thf(4513,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)],[1001]) ).

thf(364,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__mono_H) ).

thf(2349,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[364]) ).

thf(1084,axiom,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(4736,plain,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).

thf(726,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__ring__abs @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__ring__abs) ).

thf(3573,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)],[726]) ).

thf(869,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__irrefl__nat) ).

thf(4049,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(1044,axiom,
    ! [A: $i,B: $i] :
      ( ( ( class_Power_Opower @ B )
        & ( class_Rings_Osemiring__0 @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( c_Nat_OSuc @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__0__Suc) ).

thf(4653,plain,
    ! [A: $i,B: $i] :
      ( ( ( class_Power_Opower @ B )
        & ( class_Rings_Osemiring__0 @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( c_Nat_OSuc @ A ) )
        = ( c_Groups_Ozero__class_Ozero @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(598,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__antisym) ).

thf(3141,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(648,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/sandbox2/benchmark/theBenchmark.p',fact_diff__cancel) ).

thf(3311,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)],[648]) ).

thf(518,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/sandbox2/benchmark/theBenchmark.p',fact_add__0__right) ).

thf(2877,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)],[518]) ).

thf(1239,axiom,
    class_Groups_Ozero @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ozero) ).

thf(4983,plain,
    class_Groups_Ozero @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1239]) ).

thf(802,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) ).

thf(3846,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)],[802]) ).

thf(58,axiom,
    class_Groups_Oordered__ab__group__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__group__add) ).

thf(1424,plain,
    class_Groups_Oordered__ab__group__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(397,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( c_RComplete_Onatfloor @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__natfloor) ).

thf(2455,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[397]) ).

thf(1011,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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Ozero) ).

thf(4545,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)],[1011]) ).

thf(1216,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/sandbox2/benchmark/theBenchmark.p',fact_Re_Odiff) ).

thf(4938,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)],[1216]) ).

thf(53,axiom,
    class_Groups_Oab__group__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__group__add) ).

thf(1414,plain,
    class_Groups_Oab__group__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(329,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__diff__assoc2) ).

thf(2240,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[329]) ).

thf(204,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Opreorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Opreorder) ).

thf(1842,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Opreorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(1090,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4744,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).

thf(161,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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Oadd) ).

thf(1700,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)],[161]) ).

thf(192,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( C = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I4_J) ).

thf(1810,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( C = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(303,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_decr__mult__lemma) ).

thf(2172,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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)],[303]) ).

thf(422,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/sandbox2/benchmark/theBenchmark.p',fact_add__scale__eq__noteq) ).

thf(2551,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)],[422]) ).

thf(280,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__cancel__right) ).

thf(2085,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[280]) ).

thf(770,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Odiff) ).

thf(3723,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)],[770]) ).

thf(1167,axiom,
    class_Groups_Oordered__ab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add) ).

thf(4867,plain,
    class_Groups_Oordered__ab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).

thf(220,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__ge__self) ).

thf(1889,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(719,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__trans) ).

thf(3555,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(1133,axiom,
    class_Power_Opower @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Power_Opower) ).

thf(4814,plain,
    class_Power_Opower @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).

thf(1125,axiom,
    class_Groups_Oab__group__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__group__add) ).

thf(4804,plain,
    class_Groups_Oab__group__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).

thf(26,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__zero__iff) ).

thf(1324,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)],[26]) ).

thf(184,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/sandbox2/benchmark/theBenchmark.p',fact_K__record__comp) ).

thf(1776,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)],[184]) ).

thf(412,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) )
      <=> ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__mult__iff) ).

thf(2519,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) )
         => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ) )
        & ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[412]) ).

thf(15,axiom,
    class_Groups_Oab__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__add) ).

thf(1297,plain,
    class_Groups_Oab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(437,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_monoseq__realpow) ).

thf(2598,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[437]) ).

thf(330,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__ab__semigroup__add__imp__le @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(2243,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)],[330]) ).

thf(932,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
      <=> ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
           => ? [D: $i] :
              ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ E ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ F ) )
                   => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Cauchy__iff) ).

thf(4244,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OCauchy @ B @ A )
         => ! [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
             => ? [D: $i] :
                ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ E ) )
                 => ! [F: $i] :
                      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ F ) )
                     => ( hBOOL @ ( hAPP @ ( 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] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
             => ? [D: $i] :
                ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ E ) )
                 => ! [F: $i] :
                      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ F ) )
                     => ( hBOOL @ ( hAPP @ ( 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)],[932]) ).

thf(1214,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/sandbox2/benchmark/theBenchmark.p',fact_Re_Oadd) ).

thf(4934,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)],[1214]) ).

thf(892,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( B != A )
       => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE) ).

thf(4120,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( B != A )
       => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(737,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__zero) ).

thf(3628,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ A ) @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[737]) ).

thf(900,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natceiling__eq) ).

thf(4152,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[900]) ).

thf(296,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_incr__mult__lemma) ).

thf(2140,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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)],[296]) ).

thf(926,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) ).

thf(4223,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)],[926]) ).

thf(105,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__SucI) ).

thf(1539,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(507,axiom,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__eq__plus1__left) ).

thf(2836,plain,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(68,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( ( c_Nat_OSuc @ B )
         != A )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__lessI) ).

thf(1456,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( ( c_Nat_OSuc @ B )
         != A )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(153,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/sandbox2/benchmark/theBenchmark.p',fact_mult__zero__left) ).

thf(1668,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)],[153]) ).

thf(250,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel__le) ).

thf(1974,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[250]) ).

thf(794,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/sandbox2/benchmark/theBenchmark.p',fact_neg__equal__iff__equal) ).

thf(3819,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)],[794]) ).

thf(674,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__zeroE) ).

thf(3390,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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)],[674]) ).

thf(1143,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oab__semigroup__mult) ).

thf(4829,plain,
    class_Groups_Oab__semigroup__mult @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).

thf(832,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/sandbox2/benchmark/theBenchmark.p',fact_zmult__1__right) ).

thf(3937,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)],[832]) ).

thf(1077,axiom,
    class_Rings_Oring__1__no__zero__divisors @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors) ).

thf(4717,plain,
    class_Rings_Oring__1__no__zero__divisors @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).

thf(208,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natfloor__subtract) ).

thf(1852,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[208]) ).

thf(499,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/sandbox2/benchmark/theBenchmark.p',fact_add__0__iff) ).

thf(2813,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)],[499]) ).

thf(483,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ ( hAPP @ B @ C ) ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__val__lemma) ).

thf(2766,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ ( hAPP @ B @ C ) ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(758,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/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib) ).

thf(3692,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)],[758]) ).

thf(402,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__diff__iff) ).

thf(2482,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[402]) ).

thf(1188,axiom,
    class_Rings_Olinordered__ring__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__ring__strict) ).

thf(4902,plain,
    class_Rings_Olinordered__ring__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).

thf(251,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq2) ).

thf(1979,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[251]) ).

thf(578,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__diff) ).

thf(3074,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[578]) ).

thf(875,axiom,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Im_Opos__bounded) ).

thf(4069,plain,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[875]) ).

thf(555,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/sandbox2/benchmark/theBenchmark.p',fact_abs__eq__0) ).

thf(3005,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)],[555]) ).

thf(972,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) )
         => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leE) ).

thf(4416,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) )
         => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(665,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/sandbox2/benchmark/theBenchmark.p',fact_power__abs) ).

thf(3364,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)],[665]) ).

thf(139,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/sandbox2/benchmark/theBenchmark.p',fact_zminus__zminus) ).

thf(1627,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)],[139]) ).

thf(372,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__less__mono) ).

thf(2370,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[372]) ).

thf(619,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__less__mono2) ).

thf(3215,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[619]) ).

thf(600,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel1) ).

thf(3147,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[600]) ).

thf(698,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__cancel__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring) ).

thf(3476,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)],[698]) ).

thf(1009,axiom,
    ( ( c_Complex_Oexpi @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_expi__zero) ).

thf(4535,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)],[1009]) ).

thf(256,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/sandbox2/benchmark/theBenchmark.p',fact_equal__neg__zero) ).

thf(1991,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)],[256]) ).

thf(193,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__quotient__aux1) ).

thf(1814,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[193]) ).

thf(185,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/sandbox2/benchmark/theBenchmark.p',fact_power__eq__if) ).

thf(1779,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)],[185]) ).

thf(316,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__refl) ).

thf(2208,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(805,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/sandbox2/benchmark/theBenchmark.p',fact_expand__complex__eq) ).

thf(3856,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)],[805]) ).

thf(478,axiom,
    ! [A: $i] :
      ( ( c_Complex_Ocomplex_OComplex @ ( hAPP @ c_Complex_ORe @ A ) @ ( hAPP @ c_Complex_OIm @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__surj) ).

thf(2748,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)],[478]) ).

thf(851,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) ).

thf(3999,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)],[851]) ).

thf(151,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Opos__bounded) ).

thf(1663,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[151]) ).

thf(459,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__eq__add__iff2) ).

thf(2662,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[459]) ).

thf(359,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ A @ ( c_Groups_Ozero__class_Ozero @ A ) ) @ ( c_Groups_Oone__class_Oone @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__one) ).

thf(2338,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( hBOOL @ ( hAPP @ ( 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)],[359]) ).

thf(389,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__less__abs__iff) ).

thf(2423,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[389]) ).

thf(1247,axiom,
    class_Rings_Olinordered__semiring__1__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__1__strict) ).

thf(4995,plain,
    class_Rings_Olinordered__semiring__1__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1247]) ).

thf(1013,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__realpow) ).

thf(4550,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[1013]) ).

thf(448,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ouminus @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ouminus) ).

thf(2634,plain,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ouminus @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(820,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__class_Odistrib) ).

thf(3903,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)],[820]) ).

thf(958,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__left__mono) ).

thf(4352,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[958]) ).

thf(314,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_sgn__neg) ).

thf(2203,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[314]) ).

thf(467,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I9_J) ).

thf(2702,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(722,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B ) @ A ) )
      <=> ! [E: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ B @ E ) ) @ ( hAPP @ A @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__fun__def) ).

thf(3563,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B ) @ A ) )
         => ! [E: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ B @ E ) ) @ ( hAPP @ A @ E ) ) ) )
        & ( ! [E: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( hAPP @ B @ E ) ) @ ( hAPP @ A @ E ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(271,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr__implies__not0) ).

thf(2043,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[271]) ).

thf(1183,axiom,
    class_RealVector_Ometric__space @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Ometric__space) ).

thf(4889,plain,
    class_RealVector_Ometric__space @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).

thf(729,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_leI) ).

thf(3587,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(1230,axiom,
    class_Groups_Omonoid__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__add) ).

thf(4966,plain,
    class_Groups_Omonoid__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1230]) ).

thf(1100,axiom,
    class_Orderings_Opreorder @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Opreorder) ).

thf(4765,plain,
    class_Orderings_Opreorder @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).

thf(164,axiom,
    ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re_Ozero) ).

thf(1712,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)],[164]) ).

thf(731,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/sandbox2/benchmark/theBenchmark.p',fact_add__left__cancel) ).

thf(3592,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)],[731]) ).

thf(206,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( A != B )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I11_J) ).

thf(1846,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( A != B )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(1192,axiom,
    class_Orderings_Oord @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Oord) ).

thf(4906,plain,
    class_Orderings_Oord @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).

thf(741,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__strict__increasing) ).

thf(3639,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[741]) ).

thf(394,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/sandbox2/benchmark/theBenchmark.p',fact_norm__eq__zero) ).

thf(2440,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)],[394]) ).

thf(91,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(1515,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(322,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/sandbox2/benchmark/theBenchmark.p',fact_add__diff__cancel) ).

thf(2221,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)],[322]) ).

thf(308,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( B = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__le__eq__trans) ).

thf(2189,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( B = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(136,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc__eq) ).

thf(1616,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(1206,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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__monoseq__convergent) ).

thf(4921,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)],[1206]) ).

thf(538,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) ).

thf(2936,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)],[538]) ).

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

thf(3660,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(1204,axiom,
    class_Rings_Oring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__1) ).

thf(4919,plain,
    class_Rings_Oring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1204]) ).

thf(895,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_COMBK @ D @ C @ B ) @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBK__1) ).

thf(4131,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)],[895]) ).

thf(1070,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/sandbox2/benchmark/theBenchmark.p',fact_Im_Ominus) ).

thf(4708,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)],[1070]) ).

thf(988,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__cube) ).

thf(4470,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[988]) ).

thf(1030,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__le__eq__diff) ).

thf(4613,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(1025,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/sandbox2/benchmark/theBenchmark.p',fact_zminus__zadd__distrib) ).

thf(4594,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)],[1025]) ).

thf(75,axiom,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(1483,plain,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(846,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Ocomm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocomm__monoid__add) ).

thf(3985,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)],[846]) ).

thf(633,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__gt__zero__iff) ).

thf(3262,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[633]) ).

thf(676,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__add__eq1) ).

thf(3396,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[676]) ).

thf(293,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/sandbox2/benchmark/theBenchmark.p',fact_mult__cancel1) ).

thf(2121,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)],[293]) ).

thf(327,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__comm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__comm__monoid__add) ).

thf(2235,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)],[327]) ).

thf(764,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__ge__zero) ).

thf(3707,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[764]) ).

thf(89,axiom,
    class_Orderings_Oorder @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Oorder) ).

thf(1510,plain,
    class_Orderings_Oorder @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(380,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ E )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__diff__triangle__ineq) ).

thf(2391,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ E )
     => ( hBOOL @ ( hAPP @ ( 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)],[380]) ).

thf(528,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/sandbox2/benchmark/theBenchmark.p',fact_Complex__eq__i) ).

thf(2904,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)],[528]) ).

thf(244,axiom,
    ? [A: $i] :
    ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Obounded) ).

thf(1959,plain,
    ? [A: $i] :
    ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ 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)],[244]) ).

thf(890,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) ) @ ( hAPP @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__funE) ).

thf(4116,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Orderings_Oord @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ E ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( hAPP @ C @ A ) ) @ ( hAPP @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(959,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/sandbox2/benchmark/theBenchmark.p',fact_minus__mult__right) ).

thf(4354,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)],[959]) ).

thf(579,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/sandbox2/benchmark/theBenchmark.p',fact_minus__mult__commute) ).

thf(3079,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)],[579]) ).

thf(428,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zadd__left__mono) ).

thf(2570,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[428]) ).

thf(934,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Oone @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oone) ).

thf(4256,plain,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Oone @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(540,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__0__eq) ).

thf(2941,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[540]) ).

thf(1122,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OBseq @ tc_RealDef_Oreal @ A )
     => ( ! [B: $i,C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__mono__convergent) ).

thf(4797,plain,
    ! [A: $i] :
      ( ( c_SEQ_OBseq @ tc_RealDef_Oreal @ A )
     => ( ! [B: $i,C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[1122]) ).

thf(439,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/sandbox2/benchmark/theBenchmark.p',fact_minus__add__distrib) ).

thf(2602,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)],[439]) ).

thf(17,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
      | ( B = A )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__linear) ).

thf(1301,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
      | ( B = A )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(127,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__0__0) ).

thf(1595,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)],[127]) ).

thf(799,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ B ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Odiff) ).

thf(3839,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ B @ A ) )
      = ( c_Groups_Ominus__class_Ominus @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ B ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(475,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__not__le) ).

thf(2734,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

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/sandbox2/benchmark/theBenchmark.p',fact_convergent__minus__iff) ).

thf(5016,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(1019,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_ln__mult) ).

thf(4577,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[1019]) ).

thf(787,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__le__zpower__abs) ).

thf(3792,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[787]) ).

thf(961,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__gr__0) ).

thf(4359,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[961]) ).

thf(635,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Oconvergent) ).

thf(3274,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)],[635]) ).

thf(938,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_ln__add__one__self__le__self) ).

thf(4269,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[938]) ).

thf(858,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__iff) ).

thf(4017,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[858]) ).

thf(524,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__idom @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__idom) ).

thf(2894,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__idom @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(42,axiom,
    class_Rings_Oordered__comm__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__comm__semiring) ).

thf(1374,plain,
    class_Rings_Oordered__comm__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(659,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Oincseq @ B @ A )
      <=> ! [C: $i,D: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_incseq__def) ).

thf(3343,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( ( c_SEQ_Oincseq @ B @ A )
         => ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ D ) ) ) ) )
        & ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[659]) ).

thf(337,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__of__neg) ).

thf(2260,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[337]) ).

thf(1114,axiom,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_OIm,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_OisUCont) ).

thf(4786,plain,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_RealDef_Oreal @ c_Complex_OIm,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).

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

thf(4998,plain,
    class_Orderings_Opreorder @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1250]) ).

thf(200,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/sandbox2/benchmark/theBenchmark.p',fact_diff__self) ).

thf(1833,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)],[200]) ).

thf(955,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__0__le__iff__le) ).

thf(4341,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[955]) ).

thf(780,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/sandbox2/benchmark/theBenchmark.p',fact_power__0) ).

thf(3768,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)],[780]) ).

thf(491,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__diff__inverse) ).

thf(2791,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( 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)],[491]) ).

thf(490,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__leE) ).

thf(2789,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(457,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__mult) ).

thf(2657,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)],[457]) ).

thf(696,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__triangle__ineq3) ).

thf(3472,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[696]) ).

thf(381,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/sandbox2/benchmark/theBenchmark.p',fact_equation__minus__iff) ).

thf(2393,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)],[381]) ).

thf(1124,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/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__mult) ).

thf(4801,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)],[1124]) ).

thf(864,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/sandbox2/benchmark/theBenchmark.p',fact_neg__equal__0__iff__equal) ).

thf(4032,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)],[864]) ).

thf(986,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/sandbox2/benchmark/theBenchmark.p',fact_expi__add) ).

thf(4465,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)],[986]) ).

thf(189,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__nonneg__eq__0__iff) ).

thf(1795,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( 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)],[189]) ).

thf(472,axiom,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Im_Ononneg__bounded) ).

thf(2716,plain,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[472]) ).

thf(1265,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/sandbox2/benchmark/theBenchmark.p',fact_Im_Oconvergent) ).

thf(5022,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)],[1265]) ).

thf(740,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( class_RealVector_Oreal__normed__vector @ D )
        & ( class_Orderings_Oord @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( 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] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_lemmaCauchy) ).

thf(3636,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( class_RealVector_Oreal__normed__vector @ D )
        & ( class_Orderings_Oord @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( 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] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[740]) ).

thf(272,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__iff__diff__le__0) ).

thf(2048,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(942,axiom,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__real__def) ).

thf(4283,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[942]) ).

thf(135,axiom,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(1615,plain,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(878,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( B != A )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__neq__implies__less) ).

thf(4077,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( B != A )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(630,axiom,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__eq__plus1) ).

thf(3255,plain,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(1208,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__comm__monoid__add) ).

thf(4926,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1208]) ).

thf(462,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Rings_Ozero__neq__one @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ozero__neq__one) ).

thf(2674,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)],[462]) ).

thf(427,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I4_J) ).

thf(2568,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[427]) ).

thf(102,axiom,
    hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_int__0__less__1) ).

thf(1535,plain,
    hBOOL @ ( hAPP @ ( 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)],[102]) ).

thf(550,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/sandbox2/benchmark/theBenchmark.p',fact_eq__add__iff1) ).

thf(2991,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)],[550]) ).

thf(1233,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring__1) ).

thf(4975,plain,
    class_Rings_Ocomm__semiring__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1233]) ).

thf(679,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I3_J) ).

thf(3404,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[679]) ).

thf(1096,axiom,
    ( ( hAPP @ c_Complex_OIm @ c_Complex_Oii )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Im__i) ).

thf(4758,plain,
    ( ( hAPP @ c_Complex_OIm @ c_Complex_Oii )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).

thf(1104,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Ometric__space @ B )
     => ( c_SEQ_Oconvergent @ B @ ( c_COMBK @ B @ tc_Nat_Onat @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_convergent__const) ).

thf(4769,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)],[1104]) ).

thf(299,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__eq__less__trans) ).

thf(2161,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(1010,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_sum__squares__le__zero__iff) ).

thf(4537,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[1010]) ).

thf(23,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__0) ).

thf(1312,plain,
    class_Rings_Ocomm__semiring__0 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(142,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/sandbox2/benchmark/theBenchmark.p',fact_zminus__0) ).

thf(1635,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)],[142]) ).

thf(971,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__right) ).

thf(4413,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[971]) ).

thf(593,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/sandbox2/benchmark/theBenchmark.p',fact_eq__neg__iff__add__eq__0) ).

thf(3123,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)],[593]) ).

thf(687,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/sandbox2/benchmark/theBenchmark.p',fact_complex__mult__def) ).

thf(3436,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)],[687]) ).

thf(700,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__one) ).

thf(3487,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)],[700]) ).

thf(735,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/sandbox2/benchmark/theBenchmark.p',fact_neg__0__equal__iff__equal) ).

thf(3619,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)],[735]) ).

thf(1181,axiom,
    class_Rings_Oring__1__no__zero__divisors @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring__1__no__zero__divisors) ).

thf(4887,plain,
    class_Rings_Oring__1__no__zero__divisors @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).

thf(309,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oabs__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oabs__if) ).

thf(2193,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oabs__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(561,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/sandbox2/benchmark/theBenchmark.p',fact_real__mult__right__cancel) ).

thf(3023,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)],[561]) ).

thf(268,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/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__if) ).

thf(2029,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)],[268]) ).

thf(913,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/sandbox2/benchmark/theBenchmark.p',fact_abs__power__minus) ).

thf(4186,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)],[913]) ).

thf(885,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/sandbox2/benchmark/theBenchmark.p',fact_real__mult__commute) ).

thf(4096,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)],[885]) ).

thf(346,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/sandbox2/benchmark/theBenchmark.p',fact_real__add__minus__iff) ).

thf(2293,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)],[346]) ).

thf(569,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__mono) ).

thf(3051,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Oordered__semiring @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[569]) ).

thf(1121,axiom,
    class_Rings_Omult__zero @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Omult__zero) ).

thf(4796,plain,
    class_Rings_Omult__zero @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).

thf(4,axiom,
    class_Rings_Oordered__cancel__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__cancel__semiring) ).

thf(1271,plain,
    class_Rings_Oordered__cancel__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(1149,axiom,
    class_Orderings_Oorder @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Oorder) ).

thf(4837,plain,
    class_Orderings_Oorder @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).

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/sandbox2/benchmark/theBenchmark.p',fact_Re_Ominus) ).

thf(5011,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(771,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/sandbox2/benchmark/theBenchmark.p',fact_eq__iff__diff__eq__0) ).

thf(3726,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)],[771]) ).

thf(744,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( 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] :
                ( ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_realpow__pos__nth__unique) ).

thf(3651,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( 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] :
                ( ( ( hBOOL @ ( hAPP @ ( 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)],[744]) ).

thf(759,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__diff__ineq) ).

thf(3695,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[759]) ).

thf(886,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq2) ).

thf(4099,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[886]) ).

thf(654,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__imp__le__right) ).

thf(3330,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(908,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__refl) ).

thf(4171,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(212,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I8_J) ).

thf(1865,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(658,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/sandbox2/benchmark/theBenchmark.p',fact_norm__one) ).

thf(3340,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)],[658]) ).

thf(257,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/sandbox2/benchmark/theBenchmark.p',fact_mult__0) ).

thf(1997,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)],[257]) ).

thf(761,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__increasing2) ).

thf(3700,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[761]) ).

thf(873,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neq0__conv) ).

thf(4060,plain,
    ! [A: $i] :
      ( ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( 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)],[873]) ).

thf(398,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/sandbox2/benchmark/theBenchmark.p',fact_power__one__right) ).

thf(2457,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)],[398]) ).

thf(1161,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/sandbox2/benchmark/theBenchmark.p',fact_decseq__eq__incseq) ).

thf(4854,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)],[1161]) ).

thf(742,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/sandbox2/benchmark/theBenchmark.p',fact_combine__common__factor) ).

thf(3641,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)],[742]) ).

thf(368,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) ).

thf(2358,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)],[368]) ).

thf(976,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I3_J) ).

thf(4433,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(37,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_less__bin__lemma) ).

thf(1356,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(1076,axiom,
    class_Rings_Oordered__comm__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__comm__semiring) ).

thf(4716,plain,
    class_Rings_Oordered__comm__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(1117,axiom,
    class_Groups_Ominus @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ominus) ).

thf(4790,plain,
    class_Groups_Ominus @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).

thf(1211,axiom,
    class_Orderings_Olinorder @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Olinorder) ).

thf(4929,plain,
    class_Orderings_Olinorder @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1211]) ).

thf(319,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Oabs__class_Oabs @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__not__less__zero) ).

thf(2214,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ~ ( hBOOL @ ( hAPP @ ( 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)],[319]) ).

thf(1142,axiom,
    class_RealVector_Oreal__normed__vector @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__vector) ).

thf(4828,plain,
    class_RealVector_Oreal__normed__vector @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).

thf(82,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oboolean__algebra @ B )
     => ( class_Lattices_Oboolean__algebra @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Lattices_Oboolean__algebra) ).

thf(1492,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)],[82]) ).

thf(1225,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/sandbox2/benchmark/theBenchmark.p',fact_diff__commute) ).

thf(4958,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)],[1225]) ).

thf(613,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Ononneg__bounded) ).

thf(3197,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ B )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
          & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[613]) ).

thf(495,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_sgn__pos) ).

thf(2802,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[495]) ).

thf(944,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_q__pos__lemma) ).

thf(4300,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ A ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[944]) ).

thf(525,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Osgn__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Osgn__if) ).

thf(2896,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Osgn__if @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(672,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/sandbox2/benchmark/theBenchmark.p',fact_odd__nonzero) ).

thf(3384,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)],[672]) ).

thf(278,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__double__add__iff__zero__less__single__add) ).

thf(2079,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[278]) ).

thf(369,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_convergent__realpow) ).

thf(2361,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[369]) ).

thf(464,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__imp__diff__is__add) ).

thf(2679,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[464]) ).

thf(1066,axiom,
    class_Groups_Oabs__if @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oabs__if) ).

thf(4704,plain,
    class_Groups_Oabs__if @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(628,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ ( c_Groups_Oone__class_Oone @ D ) ) )
           => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__strict__decreasing) ).

thf(3251,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ ( c_Groups_Oone__class_Oone @ D ) ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[628]) ).

thf(963,axiom,
    ! [A: $i] :
      ( ( c_RealVector_Onorm__class_Onorm @ tc_RealDef_Oreal @ A )
      = ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__norm__def) ).

thf(4369,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)],[963]) ).

thf(452,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
       => ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_BseqI) ).

thf(2642,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
       => ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[452]) ).

thf(792,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__gt__zero) ).

thf(3815,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[792]) ).

thf(405,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/sandbox2/benchmark/theBenchmark.p',fact_add__minus__cancel) ).

thf(2493,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)],[405]) ).

thf(548,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__le__less__imp__less) ).

thf(2987,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[548]) ).

thf(566,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/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__eq) ).

thf(3038,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)],[566]) ).

thf(946,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__le__D1) ).

thf(4305,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(1032,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) )
            & ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_ex__least__nat__less) ).

thf(4620,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) )
            & ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( 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)],[1032]) ).

thf(191,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl2) ).

thf(1806,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(1200,axiom,
    class_Rings_Olinordered__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring) ).

thf(4914,plain,
    class_Rings_Olinordered__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).

thf(1062,axiom,
    class_Orderings_Oord @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oord) ).

thf(4699,plain,
    class_Orderings_Oord @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(35,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__zero) ).

thf(1349,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)],[35]) ).

thf(634,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__eq__refl) ).

thf(3270,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(336,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__neq__le__trans) ).

thf(2257,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(772,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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_OCauchy) ).

thf(3732,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)],[772]) ).

thf(237,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__mono) ).

thf(1933,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ ( c_Groups_Ozero__class_Ozero @ E ) ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[237]) ).

thf(168,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/sandbox2/benchmark/theBenchmark.p',fact_mult__sgn__abs) ).

thf(1722,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)],[168]) ).

thf(1126,axiom,
    ( ( hAPP @ c_Complex_ORe @ ( c_Groups_Oone__class_Oone @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__one) ).

thf(4805,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)],[1126]) ).

thf(1022,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natfloor__eq) ).

thf(4586,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[1022]) ).

thf(508,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Nat_Odiff__diff__eq) ).

thf(2839,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[508]) ).

thf(120,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__semigroup__add) ).

thf(1571,plain,
    class_Groups_Ocancel__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(44,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__lessD) ).

thf(1376,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(311,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__right__mono__neg) ).

thf(2197,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[311]) ).

thf(889,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B = A )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__eq__iff) ).

thf(4107,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( ( B = A )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(248,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__le__less__mono) ).

thf(1969,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[248]) ).

thf(19,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oord @ B )
     => ( class_Orderings_Oord @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Oord) ).

thf(1307,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oord @ B )
     => ( class_Orderings_Oord @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(447,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__less__two) ).

thf(2632,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( hBOOL @ ( hAPP @ ( 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)],[447]) ).

thf(1197,axiom,
    class_Groups_Omonoid__mult @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Omonoid__mult) ).

thf(4911,plain,
    class_Groups_Omonoid__mult @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).

thf(1241,axiom,
    class_Rings_Ocomm__semiring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__semiring) ).

thf(4985,plain,
    class_Rings_Ocomm__semiring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1241]) ).

thf(920,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).

thf(4208,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)],[920]) ).

thf(611,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natfloor__add__one) ).

thf(3192,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[611]) ).

thf(202,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__mult__pos2) ).

thf(1838,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(435,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_compl__mono) ).

thf(2593,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[435]) ).

thf(841,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatfloor @ B ) ) @ ( c_RComplete_Onatfloor @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__mono) ).

thf(3971,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[841]) ).

thf(334,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono2) ).

thf(2252,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[334]) ).

thf(343,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) ).

thf(2278,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)],[343]) ).

thf(535,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__le__iff__le) ).

thf(2928,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[535]) ).

thf(546,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/sandbox2/benchmark/theBenchmark.p',fact_mult__idem) ).

thf(2974,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)],[546]) ).

thf(502,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__mult__natfloor) ).

thf(2824,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[502]) ).

thf(1057,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__semigroup__add) ).

thf(4694,plain,
    class_Groups_Ocancel__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(440,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__diff__conv) ).

thf(2605,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[440]) ).

thf(441,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__comm__semiring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict) ).

thf(2610,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)],[441]) ).

thf(571,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/sandbox2/benchmark/theBenchmark.p',fact_diff__cancel2) ).

thf(3055,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)],[571]) ).

thf(113,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__less__Suc) ).

thf(1557,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(444,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Omult__zero @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Omult__zero) ).

thf(2625,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)],[444]) ).

thf(1145,axiom,
    class_Groups_Ogroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ogroup__add) ).

thf(4833,plain,
    class_Groups_Ogroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).

thf(140,axiom,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__neq__Zero) ).

thf(1630,plain,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(1135,axiom,
    class_Groups_Oordered__ab__group__add__abs @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs) ).

thf(4818,plain,
    class_Groups_Oordered__ab__group__add__abs @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).

thf(486,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leD1) ).

thf(2780,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(340,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Ocomm__semiring__0 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__semiring__0) ).

thf(2267,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)],[340]) ).

thf(862,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono) ).

thf(4028,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[862]) ).

thf(196,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/sandbox2/benchmark/theBenchmark.p',fact_minus__mult__minus) ).

thf(1821,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)],[196]) ).

thf(171,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/sandbox2/benchmark/theBenchmark.p',fact_realpow__num__eq__if) ).

thf(1731,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)],[171]) ).

thf(789,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/sandbox2/benchmark/theBenchmark.p',fact_complex__diff__def) ).

thf(3800,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)],[789]) ).

thf(395,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/sandbox2/benchmark/theBenchmark.p',fact_norm__sgn) ).

thf(2446,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)],[395]) ).

thf(481,axiom,
    ! [A: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_not__real__square__gt__zero) ).

thf(2757,plain,
    ! [A: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( 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 ) )
       => ~ ( hBOOL @ ( hAPP @ ( 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)],[481]) ).

thf(112,axiom,
    class_Rings_Olinordered__ring__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__ring__strict) ).

thf(1556,plain,
    class_Rings_Olinordered__ring__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(70,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ ( c_Nat_OSuc @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__less__SucD) ).

thf(1460,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Nat_OSuc @ B ) ) @ ( c_Nat_OSuc @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(1054,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__neq) ).

thf(4684,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(509,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_ln__eq__zero__iff) ).

thf(2842,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[509]) ).

thf(243,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/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__cancel1) ).

thf(1952,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)],[243]) ).

thf(1244,axiom,
    class_Rings_Oidom @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oidom) ).

thf(4990,plain,
    class_Rings_Oidom @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1244]) ).

thf(158,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Oadd) ).

thf(1688,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)],[158]) ).

thf(1079,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/sandbox2/benchmark/theBenchmark.p',help_c__COMBB__1) ).

thf(4720,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)],[1079]) ).

thf(126,axiom,
    class_Rings_Oring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring) ).

thf(1594,plain,
    class_Rings_Oring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(318,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__semiring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__semiring__strict) ).

thf(2212,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)],[318]) ).

thf(145,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/sandbox2/benchmark/theBenchmark.p',fact_power__power__power) ).

thf(1643,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)],[145]) ).

thf(756,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I12_J) ).

thf(3687,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(1222,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/sandbox2/benchmark/theBenchmark.p',fact_monoseq__minus) ).

thf(4954,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)],[1222]) ).

thf(552,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_complex__mod__triangle__ineq2) ).

thf(2999,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[552]) ).

thf(275,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__nonpos__nonpos) ).

thf(2069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[275]) ).

thf(701,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__nonpos__nonpos) ).

thf(3490,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[701]) ).

thf(715,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/sandbox2/benchmark/theBenchmark.p',fact_o__assoc) ).

thf(3539,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)],[715]) ).

thf(1156,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__mult) ).

thf(4844,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).

thf(273,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/sandbox2/benchmark/theBenchmark.p',fact_square__eq__iff) ).

thf(2051,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)],[273]) ).

thf(682,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( B != A )
       => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE__linordered__idom) ).

thf(3423,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( B != A )
       => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(66,axiom,
    ! [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
    <=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) ) @ ( hAPP @ A @ ( c_Nat_OSuc @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subseq__Suc__iff) ).

thf(1445,plain,
    ! [A: $i] :
      ( ( ( c_SEQ_Osubseq @ A )
       => ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) ) @ ( hAPP @ A @ ( c_Nat_OSuc @ B ) ) ) ) )
      & ( ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[66]) ).

thf(97,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sgn__less) ).

thf(1528,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Osgn__class_Osgn @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[97]) ).

thf(881,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/sandbox2/benchmark/theBenchmark.p',fact_mult__0__right) ).

thf(4085,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)],[881]) ).

thf(603,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__le__iff) ).

thf(3159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(1141,axiom,
    c_SEQ_Oconvergent @ 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/sandbox2/benchmark/theBenchmark.p',fact_conv1) ).

thf(4827,plain,
    c_SEQ_Oconvergent @ 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)],[1141]) ).

thf(536,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/sandbox2/benchmark/theBenchmark.p',fact_abs__mult__self) ).

thf(2931,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)],[536]) ).

thf(836,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/sandbox2/benchmark/theBenchmark.p',fact_real__add__mult__distrib) ).

thf(3953,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)],[836]) ).

thf(625,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__right__mono) ).

thf(3241,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[625]) ).

thf(1203,axiom,
    class_Groups_Ouminus @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ouminus) ).

thf(4918,plain,
    class_Groups_Ouminus @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1203]) ).

thf(265,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__pos__pos) ).

thf(2022,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[265]) ).

thf(1065,axiom,
    class_Groups_Oone @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Oone) ).

thf(4703,plain,
    class_Groups_Oone @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(417,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__add1) ).

thf(2535,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[417]) ).

thf(286,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).

thf(2101,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)],[286]) ).

thf(708,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
      <=> ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
            & ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Bseq__def) ).

thf(3512,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OBseq @ B @ A )
         => ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ C ) ) ) )
        & ( ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[708]) ).

thf(608,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__imp__less__left) ).

thf(3178,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(874,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) ).

thf(4067,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[874]) ).

thf(1184,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult) ).

thf(4890,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).

thf(822,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__minus__le__zero) ).

thf(3908,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( hBOOL @ ( hAPP @ ( 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)],[822]) ).

thf(809,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__Suc__0) ).

thf(3876,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Power_Opower__class_Opower @ tc_Nat_Onat ) @ ( c_Nat_OSuc @ ( 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)],[809]) ).

thf(466,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring__abs @ C )
     => ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
            | ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__eq__mult) ).

thf(2693,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__ring__abs @ C )
     => ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
            | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
            | ( hBOOL @ ( hAPP @ ( 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)],[466]) ).

thf(1229,axiom,
    class_Rings_Osemiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Osemiring) ).

thf(4965,plain,
    class_Rings_Osemiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1229]) ).

thf(138,axiom,
    class_Groups_Oabs__if @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oabs__if) ).

thf(1626,plain,
    class_Groups_Oabs__if @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(1087,axiom,
    class_Groups_Oab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__add) ).

thf(4741,plain,
    class_Groups_Oab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).

thf(18,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Suc__pred) ).

thf(1304,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[18]) ).

thf(1168,axiom,
    class_RealVector_Ometric__space @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Ometric__space) ).

thf(4868,plain,
    class_RealVector_Ometric__space @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).

thf(641,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__left__mono) ).

thf(3293,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[641]) ).

thf(1083,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/sandbox2/benchmark/theBenchmark.p',fact_f_I2_J) ).

thf(4735,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)],[1083]) ).

thf(1176,axiom,
    class_Rings_Ocomm__ring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__ring__1) ).

thf(4879,plain,
    class_Rings_Ocomm__ring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).

thf(911,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__le__zero__iff) ).

thf(4177,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[911]) ).

thf(803,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__nonneg__nonneg) ).

thf(3849,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[803]) ).

thf(260,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__le__mult__iff) ).

thf(2005,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[260]) ).

thf(52,axiom,
    class_Groups_Ominus @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ominus) ).

thf(1413,plain,
    class_Groups_Ominus @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(974,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Ocomm__neutral) ).

thf(4426,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)],[974]) ).

thf(241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__asym) ).

thf(1948,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(1193,axiom,
    class_Rings_Oordered__ring__abs @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__ring__abs) ).

thf(4907,plain,
    class_Rings_Oordered__ring__abs @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).

thf(563,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ C @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__diff__less) ).

thf(3031,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[563]) ).

thf(124,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( class_Orderings_Oorder @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Oorder) ).

thf(1583,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( class_Orderings_Oorder @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(897,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/sandbox2/benchmark/theBenchmark.p',fact_crossproduct__eq) ).

thf(4136,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)],[897]) ).

thf(284,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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_OisUCont) ).

thf(2096,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)],[284]) ).

thf(177,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__gt1__lemma) ).

thf(1758,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[177]) ).

thf(288,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ A )
     => ( class_Rings_Ocomm__ring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ocomm__ring__1) ).

thf(2107,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)],[288]) ).

thf(1180,axiom,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(4886,plain,
    class_Groups_Oordered__ab__semigroup__add__imp__le @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).

thf(936,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/sandbox2/benchmark/theBenchmark.p',fact_double__eq__0__iff) ).

thf(4260,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)],[936]) ).

thf(1132,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add) ).

thf(4813,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).

thf(1020,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/sandbox2/benchmark/theBenchmark.p',fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) ).

thf(4580,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)],[1020]) ).

thf(1267,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ocomm__semiring__1) ).

thf(5025,plain,
    class_Rings_Ocomm__semiring__1 @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1267]) ).

thf(351,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__Re__le__cmod) ).

thf(2318,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[351]) ).

thf(488,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/sandbox2/benchmark/theBenchmark.p',fact_diff__add__cancel) ).

thf(2784,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)],[488]) ).

thf(1007,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Odiff__right) ).

thf(4529,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)],[1007]) ).

thf(269,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/sandbox2/benchmark/theBenchmark.p',fact_complex_Osimps_I2_J) ).

thf(2037,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)],[269]) ).

thf(870,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
       => ( ( c_Groups_Oabs__class_Oabs @ B @ A )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__of__pos) ).

thf(4052,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ B )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[870]) ).

thf(950,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Ominus) ).

thf(4319,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_Ocnj @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ A ) )
      = ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_Complex_Ocomplex ) @ ( hAPP @ c_Complex_Ocnj @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(568,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natfloor__add) ).

thf(3048,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[568]) ).

thf(1164,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/sandbox2/benchmark/theBenchmark.p',fact_Im_Oadd) ).

thf(4859,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)],[1164]) ).

thf(827,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__ge__zero__iff) ).

thf(3921,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[827]) ).

thf(30,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ouminus @ B )
     => ( class_Groups_Ouminus @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Groups_Ouminus) ).

thf(1340,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ouminus @ B )
     => ( class_Groups_Ouminus @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(1055,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__le__cancel1) ).

thf(4688,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[1055]) ).

thf(357,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/sandbox2/benchmark/theBenchmark.p',fact_o__apply) ).

thf(2333,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)],[357]) ).

thf(1131,axiom,
    class_Rings_Olinordered__semiring__1 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semiring__1) ).

thf(4812,plain,
    class_Rings_Olinordered__semiring__1 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).

thf(110,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/sandbox2/benchmark/theBenchmark.p',fact_zmult__assoc) ).

thf(1551,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)],[110]) ).

thf(1253,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/sandbox2/benchmark/theBenchmark.p',fact_g_I2_J) ).

thf(5003,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)],[1253]) ).

thf(1242,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_assms) ).

thf(4986,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[1242]) ).

thf(907,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_compl__le__compl__iff) ).

thf(4168,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oboolean__algebra @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[907]) ).

thf(433,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/sandbox2/benchmark/theBenchmark.p',fact_real__two__squares__add__zero__iff) ).

thf(2582,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)],[433]) ).

thf(725,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__le__mono) ).

thf(3571,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[725]) ).

thf(1046,axiom,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Opos__bounded) ).

thf(4658,plain,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ 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)],[1046]) ).

thf(109,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Nat_OSuc @ B )
        = ( c_Nat_OSuc @ A ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__inject) ).

thf(1547,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Nat_OSuc @ B )
        = ( c_Nat_OSuc @ A ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(245,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
            & ! [D: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ B @ ( hAPP @ A @ D ) ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_BseqD) ).

thf(1961,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
            & ! [D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[245]) ).

thf(1166,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/sandbox2/benchmark/theBenchmark.p',fact_complex__minus__def) ).

thf(4864,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)],[1166]) ).

thf(800,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__asym_H) ).

thf(3842,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(797,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__gt__zero__iff) ).

thf(3833,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[797]) ).

thf(594,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/sandbox2/benchmark/theBenchmark.p',fact_diff__mult__distrib2) ).

thf(3129,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)],[594]) ).

thf(505,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_realpow__minus__mult) ).

thf(2831,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Omonoid__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[505]) ).

thf(774,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/sandbox2/benchmark/theBenchmark.p',fact_add__right__imp__eq) ).

thf(3737,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)],[774]) ).

thf(181,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Oone__class_Oone @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__one__less__zero) ).

thf(1770,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ~ ( hBOOL @ ( hAPP @ ( 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)],[181]) ).

thf(669,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__cancel__right) ).

thf(3376,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[669]) ).

thf(92,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zadd__strict__right__mono) ).

thf(1516,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[92]) ).

thf(514,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/sandbox2/benchmark/theBenchmark.p',fact_add__eq__0__iff) ).

thf(2863,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)],[514]) ).

thf(1215,axiom,
    class_Groups_Ocancel__comm__monoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocancel__comm__monoid__add) ).

thf(4937,plain,
    class_Groups_Ocancel__comm__monoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1215]) ).

thf(547,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__neq__iff) ).

thf(2977,plain,
    ! [A: $i,B: $i] :
      ( ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
          | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
          | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(274,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__power__nat__eq) ).

thf(2060,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
          | ( hBOOL @ ( hAPP @ ( 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 ) )
          | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[274]) ).

thf(174,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__quotient__aux2) ).

thf(1743,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[174]) ).

thf(962,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) ).

thf(4366,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)],[962]) ).

thf(580,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).

thf(3082,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)],[580]) ).

thf(1051,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
        & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel1) ).

thf(4673,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[1051]) ).

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

thf(1566,plain,
    class_Rings_Olinordered__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(297,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__sgn__def) ).

thf(2143,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[297]) ).

thf(130,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) ).

thf(1605,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(632,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__increasing) ).

thf(3260,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[632]) ).

thf(778,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/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__1__iff) ).

thf(3754,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)],[778]) ).

thf(1220,axiom,
    ( ( hAPP @ c_Complex_OIm @ ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Im_Ozero) ).

thf(4950,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)],[1220]) ).

thf(40,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__one__le__iff__zero__less) ).

thf(1368,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[40]) ).

thf(1040,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__less__imp__less) ).

thf(4641,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(304,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__add__less__zero__iff__single__add__less__zero) ).

thf(2175,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[304]) ).

thf(501,axiom,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Re_Opos__bounded) ).

thf(2821,plain,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[501]) ).

thf(617,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__diff) ).

thf(3209,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[617]) ).

thf(1226,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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__offset) ).

thf(4961,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)],[1226]) ).

thf(572,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Ono__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Ono__zero__divisors) ).

thf(3058,plain,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Ono__zero__divisors @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(385,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__gt__zero__cancel__iff) ).

thf(2407,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[385]) ).

thf(785,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_BseqI2_H) ).

thf(3786,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ D )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[785]) ).

thf(98,axiom,
    class_Rings_Oring__1__no__zero__divisors @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1__no__zero__divisors) ).

thf(1531,plain,
    class_Rings_Oring__1__no__zero__divisors @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(952,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__less__zpower__abs__iff) ).

thf(4324,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[952]) ).

thf(511,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/sandbox2/benchmark/theBenchmark.p',fact_real__add__eq__0__iff) ).

thf(2850,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)],[511]) ).

thf(76,axiom,
    class_Orderings_Opreorder @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Orderings_Opreorder) ).

thf(1484,plain,
    class_Orderings_Opreorder @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(85,axiom,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) ).

thf(1499,plain,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(977,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff__inverse) ).

thf(4437,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[977]) ).

thf(1015,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__diff__assoc) ).

thf(4561,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[1015]) ).

thf(1191,axiom,
    class_Rings_Ozero__neq__one @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ozero__neq__one) ).

thf(4905,plain,
    class_Rings_Ozero__neq__one @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).

thf(532,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ozero @ A )
     => ( class_Groups_Ozero @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ozero) ).

thf(2920,plain,
    ! [A: $i] :
      ( ( class_Groups_Ozero @ A )
     => ( class_Groups_Ozero @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(898,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__le__imp__le__exp) ).

thf(4145,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(342,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__mono) ).

thf(2276,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[342]) ).

thf(199,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__zmult__distrib2) ).

thf(1830,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)],[199]) ).

thf(224,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__sgn__pos) ).

thf(1901,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[224]) ).

thf(905,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__irrefl) ).

thf(4164,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(690,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__semiring) ).

thf(3448,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__semiring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(948,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_less__add__iff1) ).

thf(4314,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[948]) ).

thf(166,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/sandbox2/benchmark/theBenchmark.p',fact_power__add) ).

thf(1716,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)],[166]) ).

thf(833,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__le__power__abs) ).

thf(3940,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[833]) ).

thf(323,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/sandbox2/benchmark/theBenchmark.p',fact_power__real__of__nat) ).

thf(2224,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)],[323]) ).

thf(646,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/sandbox2/benchmark/theBenchmark.p',fact_Nat_Oadd__0__right) ).

thf(3305,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)],[646]) ).

thf(62,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__mult) ).

thf(1432,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(300,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Odiff__left) ).

thf(2165,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)],[300]) ).

thf(143,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = ( hAPP @ c_Complex_ORe @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__Re__cnj) ).

thf(1637,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( hAPP @ c_Complex_Ocnj @ A ) )
      = ( hAPP @ c_Complex_ORe @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(589,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Olinordered__ring__strict @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Olinordered__ring__strict) ).

thf(3114,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)],[589]) ).

thf(1261,axiom,
    class_RealVector_Oreal__normed__div__algebra @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra) ).

thf(5015,plain,
    class_RealVector_Oreal__normed__div__algebra @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1261]) ).

thf(859,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__le__power) ).

thf(4020,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[859]) ).

thf(390,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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__power) ).

thf(2429,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)],[390]) ).

thf(1103,axiom,
    class_Groups_Ocancel__semigroup__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add) ).

thf(4768,plain,
    class_Groups_Ocancel__semigroup__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).

thf(796,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__le__real__less) ).

thf(3828,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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 ) ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(353,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/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__self__implies__10) ).

thf(2322,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)],[353]) ).

thf(104,axiom,
    class_Rings_Oidom @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oidom) ).

thf(1538,plain,
    class_Rings_Oidom @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(931,axiom,
    ! [A: $i,B: $i] :
      ( ( class_SEQ_Ocomplete__space @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
      <=> ( c_SEQ_Oconvergent @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Cauchy__convergent__iff) ).

thf(4241,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)],[931]) ).

thf(420,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff__inverse2) ).

thf(2545,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[420]) ).

thf(1123,axiom,
    class_Orderings_Olinorder @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Orderings_Olinorder) ).

thf(4800,plain,
    class_Orderings_Olinorder @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).

thf(1238,axiom,
    class_Rings_Oring__no__zero__divisors @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors) ).

thf(4982,plain,
    class_Rings_Oring__no__zero__divisors @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1238]) ).

thf(681,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_sgn__real__def) ).

thf(3412,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[681]) ).

thf(1111,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/sandbox2/benchmark/theBenchmark.p',fact__096EX_Af_O_Asubseq_Af_A_G_Amonoseq_A_I_Fn_O_A_IRe_Ao_As_J_A_If_An_J_J_096) ).

thf(4781,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)],[1111]) ).

thf(663,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_OCauchy) ).

thf(3359,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)],[663]) ).

thf(1085,axiom,
    class_Groups_Ocomm__monoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocomm__monoid__add) ).

thf(4737,plain,
    class_Groups_Ocomm__monoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).

thf(10,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__less) ).

thf(1285,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[10]) ).

thf(585,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__less__real__le) ).

thf(3095,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(107,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/sandbox2/benchmark/theBenchmark.p',fact_zmult__commute) ).

thf(1543,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)],[107]) ).

thf(207,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).

thf(1849,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)],[207]) ).

thf(9,axiom,
    class_Rings_Oordered__semiring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__semiring) ).

thf(1284,plain,
    class_Rings_Oordered__semiring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(1093,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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__zero__iff) ).

thf(4748,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)],[1093]) ).

thf(432,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/sandbox2/benchmark/theBenchmark.p',fact_diff__add__inverse) ).

thf(2579,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)],[432]) ).

thf(673,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__nonpos__nonneg) ).

thf(3388,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[673]) ).

thf(125,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__add1__eq) ).

thf(1585,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[125]) ).

thf(1058,axiom,
    class_Rings_Oordered__semiring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oordered__semiring) ).

thf(4695,plain,
    class_Rings_Oordered__semiring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(474,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__0__less__mult__iff) ).

thf(2728,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[474]) ).

thf(539,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_decr__lemma) ).

thf(2939,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[539]) ).

thf(577,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ A @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__self__eq__0) ).

thf(3071,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)],[577]) ).

thf(1002,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_leD) ).

thf(4516,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(541,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( B != A )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
          | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neq__iff) ).

thf(2948,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( ( B != A )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
            | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
            | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) )
         => ( B != A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(760,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natceiling__add__one) ).

thf(3697,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[760]) ).

thf(103,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( class_Orderings_Opreorder @ ( tc_fun @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun__Orderings_Opreorder) ).

thf(1536,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Opreorder @ B )
     => ( class_Orderings_Opreorder @ ( tc_fun @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(252,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__imp__neg__le) ).

thf(1981,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[252]) ).

thf(1033,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Re) ).

thf(4624,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ c_Complex_ORe @ ( c_Complex_Ocomplex_OComplex @ B @ A ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(1029,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
         => ? [D: $i] :
            ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ E ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ F ) )
                 => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_CauchyD) ).

thf(4611,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ C )
     => ( ( c_SEQ_OCauchy @ C @ B )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
         => ? [D: $i] :
            ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ E ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ F ) )
                 => ( hBOOL @ ( hAPP @ ( 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)],[1029]) ).

thf(1234,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/sandbox2/benchmark/theBenchmark.p',fact_rcis__zero__mod) ).

thf(4976,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)],[1234]) ).

thf(77,axiom,
    class_Rings_Olinordered__semiring__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__strict) ).

thf(1485,plain,
    class_Rings_Olinordered__semiring__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(352,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__linear) ).

thf(2320,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(470,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/sandbox2/benchmark/theBenchmark.p',fact_diff__int__def) ).

thf(2711,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)],[470]) ).

thf(1245,axiom,
    class_RealVector_Oreal__normed__algebra @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__RealVector_Oreal__normed__algebra) ).

thf(4991,plain,
    class_RealVector_Oreal__normed__algebra @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1245]) ).

thf(176,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/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__0__iff) ).

thf(1749,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)],[176]) ).

thf(868,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__minus__self__iff) ).

thf(4046,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(61,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__add1__eq__le) ).

thf(1427,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[61]) ).

thf(884,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE__nat) ).

thf(4093,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(513,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__zminus__inverse2) ).

thf(2860,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)],[513]) ).

thf(1150,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Groups_Ocomm__monoid__add) ).

thf(4838,plain,
    class_Groups_Ocomm__monoid__add @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).

thf(523,axiom,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oidom) ).

thf(2892,plain,
    ! [A: $i] :
      ( ( class_Rings_Oidom @ A )
     => ( class_Rings_Oidom @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(5,axiom,
    ! [A: $i] :
      ( A
     != ( c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_n__not__Suc__n) ).

thf(1272,plain,
    ! [A: $i] :
      ( A
     != ( c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(602,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/sandbox2/benchmark/theBenchmark.p',fact_no__zero__divisors) ).

thf(3156,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)],[602]) ).

thf(710,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( B = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord__less__eq__trans) ).

thf(3520,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( B = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(510,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__Im__le__cmod) ).

thf(2848,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[510]) ).

thf(1136,axiom,
    class_Groups_Ocancel__ab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Ocancel__ab__semigroup__add) ).

thf(4819,plain,
    class_Groups_Ocancel__ab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).

thf(968,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__split) ).

thf(4392,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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_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)],[968]) ).

thf(639,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Oorder) ).

thf(3288,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Oorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(392,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/sandbox2/benchmark/theBenchmark.p',fact_Limits_Ominus__diff__minus) ).

thf(2435,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)],[392]) ).

thf(699,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__real__def) ).

thf(3478,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(728,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/sandbox2/benchmark/theBenchmark.p',fact_norm__minus__commute) ).

thf(3584,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)],[728]) ).

thf(576,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Odecseq @ B @ A )
      <=> ! [C: $i,D: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) ) @ ( hAPP @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_decseq__def) ).

thf(3067,plain,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( ( c_SEQ_Odecseq @ B @ A )
         => ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ A @ D ) ) @ ( hAPP @ A @ C ) ) ) ) )
        & ( ! [C: $i,D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( 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)],[576]) ).

thf(520,axiom,
    ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
    = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_One__nat__def) ).

thf(2885,plain,
    ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
    = ( c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(1249,axiom,
    class_Groups_Oordered__ab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) ).

thf(4997,plain,
    class_Groups_Oordered__ab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1249]) ).

thf(149,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ ( hAPP @ B @ C ) ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat0__intermed__int__val) ).

thf(1654,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ D ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ ( hAPP @ B @ C ) ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ D ) @ C ) )
              & ( ( hAPP @ B @ D )
                = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(276,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Olinorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Orderings_Olinorder) ).

thf(2071,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Orderings_Olinorder @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(517,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_one__le__power) ).

thf(2875,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[517]) ).

thf(606,axiom,
    hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).

thf(3168,plain,
    hBOOL @ ( hAPP @ ( 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)],[606]) ).

thf(1012,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__less__le__mono) ).

thf(4548,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ E @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[1012]) ).

thf(940,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) ).

thf(4278,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)],[940]) ).

thf(693,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__less__def) ).

thf(3457,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
          & ( B != A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
          & ( B != A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(25,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_pos__zmult__eq__1__iff) ).

thf(1316,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[25]) ).

thf(147,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zero__less__power) ).

thf(1649,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[147]) ).

thf(1061,axiom,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) ).

thf(4698,plain,
    class_Rings_Olinordered__comm__semiring__strict @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(401,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) ).

thf(2479,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)],[401]) ).

thf(1158,axiom,
    class_Rings_Oordered__comm__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__comm__semiring) ).

thf(4851,plain,
    class_Rings_Oordered__comm__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).

thf(678,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/sandbox2/benchmark/theBenchmark.p',fact_diff__int__def__symmetric) ).

thf(3401,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)],[678]) ).

thf(281,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_OisUCont) ).

thf(2088,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)],[281]) ).

thf(730,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/sandbox2/benchmark/theBenchmark.p',fact_fun__diff__def) ).

thf(3589,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)],[730]) ).

thf(69,axiom,
    class_Rings_Oordered__ring__abs @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oordered__ring__abs) ).

thf(1459,plain,
    class_Rings_Oordered__ring__abs @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(194,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ominus @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ominus) ).

thf(1817,plain,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ominus @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(543,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
          = ( c_Groups_Oone__class_Oone @ B ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sgn__1__pos) ).

thf(2960,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( ( c_Groups_Osgn__class_Osgn @ B @ A )
            = ( c_Groups_Oone__class_Oone @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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)],[543]) ).

thf(1218,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/sandbox2/benchmark/theBenchmark.p',fact_abs__real__of__nat__cancel) ).

thf(4944,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)],[1218]) ).

thf(1172,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/sandbox2/benchmark/theBenchmark.p',fact_Im_Odiff) ).

thf(4873,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)],[1172]) ).

thf(973,axiom,
    ! [A: $i] :
      ( ( c_SEQ_Osubseq @ A )
    <=> ! [B: $i,C: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) ) @ ( hAPP @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subseq__def) ).

thf(4419,plain,
    ! [A: $i] :
      ( ( ( c_SEQ_Osubseq @ A )
       => ! [B: $i,C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ A @ B ) ) @ ( hAPP @ A @ C ) ) ) ) )
      & ( ! [B: $i,C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[973]) ).

thf(228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__sum__triangle__ineq) ).

thf(1913,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[228]) ).

thf(1045,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__comm__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_comm__mult__left__mono) ).

thf(4656,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__comm__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[1045]) ).

thf(205,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__linear) ).

thf(1844,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Olinorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
        | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(808,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat @ B @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__self) ).

thf(3874,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[808]) ).

thf(1086,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/sandbox2/benchmark/theBenchmark.p',fact_complex__diff) ).

thf(4738,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)],[1086]) ).

thf(1110,axiom,
    class_Orderings_Oorder @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Oorder) ).

thf(4780,plain,
    class_Orderings_Oorder @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).

thf(354,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__less) ).

thf(2325,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(829,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1 @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_convex__bound__le) ).

thf(3928,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1 @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( 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 ) )
               => ( hBOOL @ ( hAPP @ ( 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)],[829]) ).

thf(736,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/sandbox2/benchmark/theBenchmark.p',fact_diffs0__imp__equal) ).

thf(3625,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)],[736]) ).

thf(1042,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__imp__le) ).

thf(4646,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(551,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le0) ).

thf(2997,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[551]) ).

thf(887,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__group__add @ A )
     => ( class_Groups_Ogroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ogroup__add) ).

thf(4102,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)],[887]) ).

thf(361,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__less__trans) ).

thf(2342,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(985,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__linear) ).

thf(4463,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(146,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) ).

thf(1646,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)],[146]) ).

thf(1199,axiom,
    class_Groups_Oone @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oone) ).

thf(4913,plain,
    class_Groups_Oone @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).

thf(285,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) ).

thf(2098,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)],[285]) ).

thf(795,axiom,
    ! [A: $i] :
      ( ( c_RComplete_Onatfloor @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natfloor__real__of__nat) ).

thf(3825,plain,
    ! [A: $i] :
      ( ( c_RComplete_Onatfloor @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(987,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__le__imp__le) ).

thf(4468,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(1053,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/sandbox2/benchmark/theBenchmark.p',fact_abs__minus__cancel) ).

thf(4681,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)],[1053]) ).

thf(954,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).

thf(4338,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)],[954]) ).

thf(537,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__triangle__ineq) ).

thf(2934,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( hBOOL @ ( hAPP @ ( 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)],[537]) ).

thf(6,axiom,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__not__Zero) ).

thf(1276,plain,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(434,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__strict__left__mono__neg) ).

thf(2591,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[434]) ).

thf(786,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/sandbox2/benchmark/theBenchmark.p',fact_minus__apply) ).

thf(3789,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)],[786]) ).

thf(765,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__0__less__iff__less) ).

thf(3709,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[765]) ).

thf(1049,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl3) ).

thf(4666,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(1257,axiom,
    ( ( c_RComplete_Onatceiling @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__zero) ).

thf(5008,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)],[1257]) ).

thf(667,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__le) ).

thf(3369,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(894,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__less__iff) ).

thf(4126,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ B ) ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[894]) ).

thf(621,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/sandbox2/benchmark/theBenchmark.p',fact_mult__is__0) ).

thf(3220,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)],[621]) ).

thf(1075,axiom,
    class_RealVector_Oreal__normed__algebra @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra) ).

thf(4715,plain,
    class_RealVector_Oreal__normed__algebra @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(1118,axiom,
    class_Rings_Ocomm__semiring__0 @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ocomm__semiring__0) ).

thf(4791,plain,
    class_Rings_Ocomm__semiring__0 @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).

thf(267,axiom,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_RealDef_Oreal ) @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_Ononneg__bounded) ).

thf(2026,plain,
    ? [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
      & ! [B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealVector_Onorm__class_Onorm @ tc_Complex_Ocomplex @ ( hAPP @ c_Complex_Ocnj @ 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)],[267]) ).

thf(738,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/sandbox2/benchmark/theBenchmark.p',fact_diff__0) ).

thf(3630,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)],[738]) ).

thf(705,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__less__imp__less__exp) ).

thf(3501,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(554,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_norm__not__less__zero) ).

thf(3003,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ~ ( hBOOL @ ( hAPP @ ( 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)],[554]) ).

thf(1258,axiom,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(5010,plain,
    class_Groups_Oordered__cancel__ab__semigroup__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1258]) ).

thf(482,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__add2) ).

thf(2764,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[482]) ).

thf(855,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocancel__comm__monoid__add @ A )
     => ( class_Groups_Ocancel__comm__monoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add) ).

thf(4010,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)],[855]) ).

thf(1146,axiom,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_Complex_Ocomplex @ c_Complex_Ocnj,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_cnj_OisUCont) ).

thf(4834,plain,
    c_Lim_OisUCont @ tc_Complex_Ocomplex @ tc_Complex_Ocomplex @ c_Complex_Ocnj,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).

thf(1067,axiom,
    class_Groups_Ominus @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ominus) ).

thf(4705,plain,
    class_Groups_Ominus @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(747,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__not__sym) ).

thf(3662,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(413,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ A @ ( c_Groups_Ozero__class_Ozero @ A ) ) @ ( c_Groups_Oone__class_Oone @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__one) ).

thf(2526,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__semidom @ A )
     => ( hBOOL @ ( hAPP @ ( 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)],[413]) ).

thf(1006,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ge__natfloor__plus__one__imp__gt) ).

thf(4527,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(388,axiom,
    ! [A: $i] :
      ( ( ( hAPP @ c_Complex_Ocnj @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_complex__cnj__zero__iff) ).

thf(2416,plain,
    ! [A: $i] :
      ( ( ( ( hAPP @ c_Complex_Ocnj @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) )
       => ( ( hAPP @ c_Complex_Ocnj @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Complex_Ocomplex ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(982,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 ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
             => ( C = A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__eq__imp__eq__base) ).

thf(4451,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 ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
           => ( ( hBOOL @ ( hAPP @ ( 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)],[982]) ).

thf(581,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__gt__zero__imp__gt__one) ).

thf(3085,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ ( c_Transcendental_Oln @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[581]) ).

thf(1091,axiom,
    ! [A: $i] :
      ( ( c_SEQ_OCauchy @ tc_RealDef_Oreal @ A )
     => ( c_SEQ_Oconvergent @ tc_RealDef_Oreal @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__Cauchy__convergent) ).

thf(4745,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)],[1091]) ).

thf(81,axiom,
    class_Rings_Osemiring__0 @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Osemiring__0) ).

thf(1491,plain,
    class_Rings_Osemiring__0 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(493,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/sandbox2/benchmark/theBenchmark.p',fact_mult__diff__mult) ).

thf(2796,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)],[493]) ).

thf(1228,axiom,
    class_Rings_Oring @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Oring) ).

thf(4964,plain,
    class_Rings_Oring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1228]) ).

thf(732,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) )
          | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel__right__disj) ).

thf(3598,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) ) ) ) )
        & ( ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) )
            | ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
              & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ C ) ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[732]) ).

thf(356,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__mult__le__cancel__iff2) ).

thf(2330,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[356]) ).

thf(132,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/sandbox2/benchmark/theBenchmark.p',fact_power_Opower_Opower__Suc) ).

thf(1607,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)],[132]) ).

thf(150,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Omonoid__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__add) ).

thf(1661,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)],[150]) ).

thf(980,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/sandbox2/benchmark/theBenchmark.p',fact_mult__1__left) ).

thf(4446,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)],[980]) ).

thf(350,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/sandbox2/benchmark/theBenchmark.p',fact_power__commutes) ).

thf(2315,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)],[350]) ).

thf(264,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__antisym) ).

thf(2019,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(916,axiom,
    hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ v_r ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rp) ).

thf(4197,plain,
    hBOOL @ ( hAPP @ ( 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)],[916]) ).

thf(652,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__eq) ).

thf(3324,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(910,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Oone__class_Oone @ C ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__right__le__one__le) ).

thf(4175,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Oone__class_Oone @ C ) ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[910]) ).

thf(810,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natceiling__subtract) ).

thf(3879,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[810]) ).

thf(922,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__mult__order) ).

thf(4214,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[922]) ).

thf(801,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__nonneg__pos) ).

thf(3844,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__comm__monoid__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[801]) ).

thf(1024,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/sandbox2/benchmark/theBenchmark.p',fact_zadd__0) ).

thf(4591,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)],[1024]) ).

thf(431,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Ominus__right) ).

thf(2576,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)],[431]) ).

thf(991,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I5_J) ).

thf(4478,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(325,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__right__mono__neg) ).

thf(2230,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[325]) ).

thf(222,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) ).

thf(1893,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)],[222]) ).

thf(20,axiom,
    class_Rings_Olinordered__semiring__1__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1__strict) ).

thf(1309,plain,
    class_Rings_Olinordered__semiring__1__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(477,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/sandbox2/benchmark/theBenchmark.p',fact_crossproduct__noteq) ).

thf(2739,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)],[477]) ).

thf(65,axiom,
    class_Groups_Ozero @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ozero) ).

thf(1444,plain,
    class_Groups_Ozero @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(64,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lessI) ).

thf(1442,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(1169,axiom,
    class_Rings_Olinordered__semidom @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__semidom) ).

thf(4869,plain,
    class_Rings_Olinordered__semidom @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).

thf(964,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      <=> ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_less__add__iff2) ).

thf(4372,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Oordered__ring @ F )
     => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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 ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[964]) ).

thf(718,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__minus__iff) ).

thf(3552,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(1209,axiom,
    class_Rings_Oring__1 @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring__1) ).

thf(4927,plain,
    class_Rings_Oring__1 @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1209]) ).

thf(670,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Orderings_Oorder @ B )
     => ( ( c_SEQ_Oincseq @ B @ A )
       => ( c_SEQ_Omonoseq @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_incseq__imp__monoseq) ).

thf(3379,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)],[670]) ).

thf(345,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__inject__exp) ).

thf(2287,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[345]) ).

thf(1202,axiom,
    class_SEQ_Ocomplete__space @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__SEQ_Ocomplete__space) ).

thf(4917,plain,
    class_SEQ_Ocomplete__space @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).

thf(183,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__trans) ).

thf(1774,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Opreorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(989,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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Ominus) ).

thf(4472,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)],[989]) ).

thf(850,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Groups_Oordered__cancel__ab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add) ).

thf(3997,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)],[850]) ).

thf(473,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/sandbox2/benchmark/theBenchmark.p',fact_power__eq__0__iff) ).

thf(2719,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)],[473]) ).

thf(815,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel__left__neg) ).

thf(3892,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__ring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ B ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[815]) ).

thf(108,axiom,
    class_Rings_Ono__zero__divisors @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ono__zero__divisors) ).

thf(1546,plain,
    class_Rings_Ono__zero__divisors @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(586,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/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__commute) ).

thf(3100,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)],[586]) ).

thf(1155,axiom,
    class_Power_Opower @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Power_Opower) ).

thf(4843,plain,
    class_Power_Opower @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).

thf(595,axiom,
    ! [A: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ A )
     => ( class_Groups_Oab__semigroup__add @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Oab__semigroup__add) ).

thf(3132,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)],[595]) ).

thf(960,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__idom @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__mult__less) ).

thf(4357,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Rings_Olinordered__idom @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ E @ ( c_Groups_Oabs__class_Oabs @ E @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[960]) ).

thf(1026,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__iff__diff__less__0) ).

thf(4597,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Ominus__class_Ominus @ C @ B @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(695,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/sandbox2/benchmark/theBenchmark.p',fact_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(3469,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)],[695]) ).

thf(745,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_abs__add__one__gt__zero) ).

thf(3658,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[745]) ).

thf(321,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/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__commute) ).

thf(2218,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)],[321]) ).

thf(374,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/sandbox2/benchmark/theBenchmark.p',fact_abs__zmult__eq__1) ).

thf(2374,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)],[374]) ).

thf(1177,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/sandbox2/benchmark/theBenchmark.p',fact_complex__Im__mult) ).

thf(4880,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)],[1177]) ).

thf(34,axiom,
    class_Rings_Olinordered__semidom @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semidom) ).

thf(1348,plain,
    class_Rings_Olinordered__semidom @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(348,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
          & ( B != A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__le) ).

thf(2302,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ( B != A ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ( B != A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(411,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/sandbox2/benchmark/theBenchmark.p',fact_mult__right_Oconvergent) ).

thf(2517,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)],[411]) ).

thf(169,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/sandbox2/benchmark/theBenchmark.p',fact_power__Suc) ).

thf(1725,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)],[169]) ).

thf(7,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) ) @ ( c_Transcendental_Oln @ A ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ln__less__cancel__iff) ).

thf(1280,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) ) @ ( c_Transcendental_Oln @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Transcendental_Oln @ B ) ) @ ( c_Transcendental_Oln @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(378,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/sandbox2/benchmark/theBenchmark.p',fact_abs__sgn) ).

thf(2385,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)],[378]) ).

thf(1068,axiom,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(4706,plain,
    class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(1050,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__le__imp__less__or__eq) ).

thf(4670,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
          | ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(1254,axiom,
    class_Groups_Oab__semigroup__mult @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__semigroup__mult) ).

thf(5004,plain,
    class_Groups_Oab__semigroup__mult @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1254]) ).

thf(1130,axiom,
    class_Groups_Omonoid__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Omonoid__add) ).

thf(4811,plain,
    class_Groups_Omonoid__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).

thf(811,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__1 @ A )
     => ( class_Groups_Omonoid__mult @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Groups_Omonoid__mult) ).

thf(3882,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)],[811]) ).

thf(48,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ D ) @ B ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__fun__def) ).

thf(1395,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oord @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ D ) @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ A ) @ B ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ D ) @ A ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(137,axiom,
    class_Groups_Oordered__comm__monoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oordered__comm__monoid__add) ).

thf(1625,plain,
    class_Groups_Oordered__comm__monoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(847,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/sandbox2/benchmark/theBenchmark.p',fact_zpower__zadd__distrib) ).

thf(3987,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)],[847]) ).

thf(1115,axiom,
    ( ( c_RComplete_Onatceiling @ ( c_Groups_Oone__class_Oone @ tc_RealDef_Oreal ) )
    = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_natceiling__one) ).

thf(4787,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)],[1115]) ).

thf(1153,axiom,
    class_Rings_Oring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Oring) ).

thf(4841,plain,
    class_Rings_Oring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).

thf(254,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__strict__mono) ).

thf(1986,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[254]) ).

thf(657,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/sandbox2/benchmark/theBenchmark.p',fact_minus__mult__left) ).

thf(3337,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)],[657]) ).

thf(155,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Oadd__right) ).

thf(1673,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)],[155]) ).

thf(1260,axiom,
    class_Groups_Oab__semigroup__add @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Oab__semigroup__add) ).

thf(5014,plain,
    class_Groups_Oab__semigroup__add @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1260]) ).

thf(757,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__left__mono__neg) ).

thf(3690,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__ring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[757]) ).

thf(74,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/sandbox2/benchmark/theBenchmark.p',fact_sgn__sgn) ).

thf(1480,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)],[74]) ).

thf(615,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( C = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I2_J) ).

thf(3203,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Orderings_Oorder @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( C = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(574,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/sandbox2/benchmark/theBenchmark.p',fact_add__mult__distrib2) ).

thf(3062,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)],[574]) ).

thf(558,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/sandbox2/benchmark/theBenchmark.p',fact_add__0__left) ).

thf(3015,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)],[558]) ).

thf(653,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_trans__le__add2) ).

thf(3328,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[653]) ).

thf(1036,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__comm__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_comm__mult__strict__left__mono) ).

thf(4631,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__comm__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[1036]) ).

thf(468,axiom,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( c_SEQ_OBseq @ B @ A )
      <=> ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
            & ? [D: $i] :
              ! [E: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Bseq__iff3) ).

thf(2704,plain,
    ! [A: $i,B: $i] :
      ( ( class_RealVector_Oreal__normed__vector @ B )
     => ( ( ( c_SEQ_OBseq @ B @ A )
         => ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ? [D: $i] :
                ! [E: $i] : ( hBOOL @ ( hAPP @ ( 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] :
              ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
              & ? [D: $i] :
                ! [E: $i] : ( hBOOL @ ( hAPP @ ( 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)],[468]) ).

thf(187,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ ( c_RComplete_Onatfloor @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__natfloor__eq) ).

thf(1789,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ ( c_RComplete_Onatfloor @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_RealDef_Oreal @ tc_Nat_Onat @ A ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ A ) @ ( c_RComplete_Onatfloor @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(12,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__antisym) ).

thf(1288,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(529,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/sandbox2/benchmark/theBenchmark.p',fact_norm__power) ).

thf(2913,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)],[529]) ).

thf(739,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ozero__neq__one @ A )
     => ( ( c_Groups_Oone__class_Oone @ A )
       != ( c_Groups_Ozero__class_Ozero @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__neq__zero) ).

thf(3633,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)],[739]) ).

thf(1198,axiom,
    class_Rings_Ozero__neq__one @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ozero__neq__one) ).

thf(4912,plain,
    class_Rings_Ozero__neq__one @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).

thf(270,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) ).

thf(2040,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)],[270]) ).

thf(567,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/sandbox2/benchmark/theBenchmark.p',fact_diff__0__right) ).

thf(3045,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)],[567]) ).

thf(845,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_natceiling__add) ).

thf(3982,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[845]) ).

thf(449,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__strict__left__mono) ).

thf(2636,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__cancel__ab__semigroup__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[449]) ).

thf(479,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/sandbox2/benchmark/theBenchmark.p',fact_abs__one) ).

thf(2751,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)],[479]) ).

thf(837,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
        <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_power__strict__increasing__iff) ).

thf(3956,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Oone__class_Oone @ D ) ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[837]) ).

thf(817,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__refl) ).

thf(3897,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(313,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__natfloor__gt__diff__one) ).

thf(2201,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[313]) ).

thf(1108,axiom,
    class_Rings_Olinordered__comm__semiring__strict @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__comm__semiring__strict) ).

thf(4777,plain,
    class_Rings_Olinordered__comm__semiring__strict @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).

thf(1094,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/sandbox2/benchmark/theBenchmark.p',fact_Re_OCauchy) ).

thf(4755,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)],[1094]) ).

thf(1237,axiom,
    class_Power_Opower @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Power_Opower) ).

thf(4981,plain,
    class_Power_Opower @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1237]) ).

thf(721,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_xt1_I5_J) ).

thf(3560,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(660,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] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_reduce__poly__simple) ).

thf(3347,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] : ( hBOOL @ ( hAPP @ ( 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)],[660]) ).

thf(307,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_ln__realpow) ).

thf(2186,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[307]) ).

thf(941,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__imp__less__right) ).

thf(4281,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(902,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).

thf(4157,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)],[902]) ).

thf(160,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__le__zero__cancel__iff) ).

thf(1693,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[160]) ).

thf(1263,axiom,
    class_Rings_Ocomm__ring @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__Rings_Ocomm__ring) ).

thf(5019,plain,
    class_Rings_Ocomm__ring @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1263]) ).

thf(861,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__le__not__le) ).

thf(4024,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Opreorder @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(429,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/sandbox2/benchmark/theBenchmark.p',fact_convergent__subseq__convergent) ).

thf(2572,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)],[429]) ).

thf(471,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( c_RComplete_Onatceiling @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__le__natceiling) ).

thf(2714,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[471]) ).

thf(683,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/sandbox2/benchmark/theBenchmark.p',fact_power__mult) ).

thf(3426,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)],[683]) ).

thf(221,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ A ) @ ( c_RealDef_Oreal @ tc_Nat_Onat @ ( c_RComplete_Onatceiling @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__natceiling__ge) ).

thf(1891,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[221]) ).

thf(360,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__ring__1 @ A )
     => ( class_Rings_Oring__1 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oring__1) ).

thf(2340,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)],[360]) ).

thf(460,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/sandbox2/benchmark/theBenchmark.p',fact_power__Suc2) ).

thf(2668,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)],[460]) ).

thf(751,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/sandbox2/benchmark/theBenchmark.p',fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).

thf(3673,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)],[751]) ).

thf(133,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__SucE) ).

thf(1610,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(1207,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/sandbox2/benchmark/theBenchmark.p',fact__096EX_Afa_O_Asubseq_Afa_A_G_Amonoseq_A_I_Fn_O_A_IIm_Ao_As_Ao_Af_J_A_Ifa_An_J_J_096) ).

thf(4923,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)],[1207]) ).

thf(198,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/sandbox2/benchmark/theBenchmark.p',fact_Complex__mult__i) ).

thf(1827,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)],[198]) ).

thf(1,conjecture,
    c_SEQ_OBseq @ 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/sandbox2/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_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(neg_conjecture,[status(cth)],[1]) ).

thf(1268,plain,
    ~ ( c_SEQ_OBseq @ 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)],[2]) ).

thf(99,axiom,
    class_Rings_Ocomm__semiring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__semiring__1) ).

thf(1532,plain,
    class_Rings_Ocomm__semiring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(45,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__less__Suc__eq) ).

thf(1378,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ ( c_Nat_OSuc @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(1173,axiom,
    class_Rings_Ono__zero__divisors @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Ono__zero__divisors) ).

thf(4876,plain,
    class_Rings_Ono__zero__divisors @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).

thf(928,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_order__less__imp__not__eq2) ).

thf(4229,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Orderings_Oorder @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
       => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(419,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__le__interval__iff) ).

thf(2539,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Oabs__class_Oabs @ tc_RealDef_Oreal @ B ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ tc_RealDef_Oreal ) @ A ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[419]) ).

thf(998,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__add__less2) ).

thf(4501,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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)],[998]) ).

thf(236,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Oprod__diff__prod) ).

thf(1930,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)],[236]) ).

thf(366,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq1) ).

thf(2353,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[366]) ).

thf(498,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
     => ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_rabs__ratiotest__lemma) ).

thf(2810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) )
     => ( ( hBOOL @ ( hAPP @ ( 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)],[498]) ).

thf(36,axiom,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osimps_I3_J) ).

thf(1352,plain,
    ! [A: $i] :
      ( ( c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(438,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Oone__class_Oone @ C ) ) )
           => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ B ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__le__one__le) ).

thf(2600,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Oone__class_Oone @ C ) ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[438]) ).

thf(526,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/sandbox2/benchmark/theBenchmark.p',fact_mult__Suc__right) ).

thf(2898,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)],[526]) ).

thf(966,axiom,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
          & ! [C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult_Ononneg__bounded) ).

thf(4382,plain,
    ! [A: $i] :
      ( ( class_RealVector_Oreal__normed__algebra @ A )
     => ? [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ B ) )
          & ! [C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( 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)],[966]) ).

thf(403,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/sandbox2/benchmark/theBenchmark.p',fact_o__def) ).

thf(2485,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)],[403]) ).

thf(46,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Nat_OSuc @ B )
        = ( c_Nat_OSuc @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Oinject) ).

thf(1384,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Nat_OSuc @ B )
          = ( c_Nat_OSuc @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( c_Nat_OSuc @ B )
          = ( c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(733,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/sandbox2/benchmark/theBenchmark.p',fact_complex_Oinject) ).

thf(3605,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)],[733]) ).

thf(123,axiom,
    class_Power_Opower @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Power_Opower) ).

thf(1582,plain,
    class_Power_Opower @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(339,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__right__mono) ).

thf(2265,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Oordered__semiring @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[339]) ).

thf(891,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__minus__mult__self__le) ).

thf(4118,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( 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)],[891]) ).

thf(793,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_real__le__trans) ).

thf(3817,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_RealDef_Oreal @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(86,axiom,
    class_Rings_Oring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Oring__1) ).

thf(1500,plain,
    class_Rings_Oring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(840,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_nat__eq__add__iff1) ).

thf(3965,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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)],[840]) ).

thf(233,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__le__imp__le) ).

thf(1923,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semiring__strict @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(970,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/sandbox2/benchmark/theBenchmark.p',fact_abs__zero) ).

thf(4410,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)],[970]) ).

thf(919,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/sandbox2/benchmark/theBenchmark.p',fact_Cauchy__subseq__Cauchy) ).

thf(4206,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)],[919]) ).

thf(234,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/sandbox2/benchmark/theBenchmark.p',fact_abs__minus__commute) ).

thf(1925,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)],[234]) ).

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

thf(4713,plain,
    class_Rings_Oordered__ring @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(775,axiom,
    ! [A: $i,B: $i] :
      ( ( class_SEQ_Ocomplete__space @ B )
     => ( ( c_SEQ_OCauchy @ B @ A )
       => ( c_SEQ_Oconvergent @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Cauchy__convergent) ).

thf(3740,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)],[775]) ).

thf(216,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) ).

thf(1880,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[216]) ).

thf(218,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__mono) ).

thf(1885,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[218]) ).

thf(1056,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/sandbox2/benchmark/theBenchmark.p',fact_DeMoivre2) ).

thf(4691,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)],[1056]) ).

thf(791,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/sandbox2/benchmark/theBenchmark.p',fact_comp__cong) ).

thf(3812,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)],[791]) ).

thf(49,axiom,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zabs__def) ).

thf(1399,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[49]) ).

thf(939,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__iff__add) ).

thf(4271,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(826,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__minus__self__iff) ).

thf(3918,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__idom @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ B @ A ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(882,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__nat__zero__code) ).

thf(4088,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( 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)],[882]) ).

thf(925,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__linear) ).

thf(4221,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(373,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__natfloor__add__one__gt) ).

thf(2372,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( 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)],[373]) ).

thf(451,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__imp__le__left) ).

thf(2640,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oordered__ab__semigroup__add__imp__le @ D )
     => ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(1034,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_add__less__mono1) ).

thf(4627,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[1034]) ).

thf(828,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
      <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__less__iff) ).

thf(3924,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__idom @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ B ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(1185,axiom,
    class_Groups_Omonoid__mult @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Groups_Omonoid__mult) ).

thf(4891,plain,
    class_Groups_Omonoid__mult @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).

thf(116,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__mult) ).

thf(1565,plain,
    class_Groups_Oab__semigroup__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(55,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zmult__zless__mono2) ).

thf(1416,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[55]) ).

thf(685,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_real__mult__less__mono2) ).

thf(3432,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ ( c_Groups_Ozero__class_Ozero @ tc_RealDef_Oreal ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_RealDef_Oreal @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[685]) ).

thf(614,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__le__self__iff) ).

thf(3200,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Olinordered__ab__group__add @ B )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ B @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ B ) @ A ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(727,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__eq__less__or__eq) ).

thf(3575,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(1082,axiom,
    class_Rings_Olinordered__idom @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Olinordered__idom) ).

thf(4734,plain,
    class_Rings_Olinordered__idom @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).

thf(819,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_one__less__power) ).

thf(3901,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Olinordered__semidom @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( c_Groups_Oone__class_Oone @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[819]) ).

thf(557,axiom,
    ! [A: $i] :
      ( ( class_Rings_Ocomm__semiring__0 @ A )
     => ( class_Rings_Osemiring__0 @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Osemiring__0) ).

thf(3013,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)],[557]) ).

thf(28,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zabs__less__one__iff) ).

thf(1331,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[28]) ).

thf(131,axiom,
    class_Rings_Ocomm__ring @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Ocomm__ring) ).

thf(1606,plain,
    class_Rings_Ocomm__ring @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(697,axiom,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Polynomial__Opoly__Rings_Oordered__ring) ).

thf(3474,plain,
    ! [A: $i] :
      ( ( class_Rings_Olinordered__idom @ A )
     => ( class_Rings_Oordered__ring @ ( tc_Polynomial_Opoly @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(480,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/sandbox2/benchmark/theBenchmark.p',fact_zmult__1) ).

thf(2754,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)],[480]) ).

thf(953,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/sandbox2/benchmark/theBenchmark.p',fact_minus__zero) ).

thf(4335,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)],[953]) ).

thf(618,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__less__iff__less) ).

thf(3212,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ C @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[618]) ).

thf(1080,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/sandbox2/benchmark/theBenchmark.p',fact_real__of__nat__1) ).

thf(4723,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)],[1080]) ).

thf(315,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_le__imp__0__less) ).

thf(2206,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[315]) ).

thf(1112,axiom,
    class_Groups_Omonoid__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Omonoid__mult) ).

thf(4784,plain,
    class_Groups_Omonoid__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).

thf(172,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/sandbox2/benchmark/theBenchmark.p',fact_diff__mult__distrib) ).

thf(1737,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)],[172]) ).

thf(521,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/sandbox2/benchmark/theBenchmark.p',fact_i__mult__Complex) ).

thf(2887,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)],[521]) ).

thf(1102,axiom,
    class_Rings_Ozero__neq__one @ tc_RealDef_Oreal,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_RealDef__Oreal__Rings_Ozero__neq__one) ).

thf(4767,plain,
    class_Rings_Ozero__neq__one @ tc_RealDef_Oreal,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).

thf(393,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I1_J) ).

thf(2438,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( 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)],[393]) ).

thf(80,axiom,
    class_Groups_Omonoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Omonoid__add) ).

thf(1490,plain,
    class_Groups_Omonoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(1088,axiom,
    class_SEQ_Ocomplete__space @ tc_Complex_Ocomplex,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Complex__Ocomplex__SEQ_Ocomplete__space) ).

thf(4742,plain,
    class_SEQ_Ocomplete__space @ tc_Complex_Ocomplex,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).

thf(1194,axiom,
    class_Rings_Oordered__semiring @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Oordered__semiring) ).

thf(4908,plain,
    class_Rings_Oordered__semiring @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).

thf(230,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__semidom @ B )
     => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_less__add__one) ).

thf(1917,plain,
    ! [A: $i,B: $i] :
      ( ( class_Rings_Olinordered__semidom @ B )
     => ( hBOOL @ ( hAPP @ ( 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)],[230]) ).

thf(650,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel2) ).

thf(3316,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( 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)],[650]) ).

thf(365,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/sandbox2/benchmark/theBenchmark.p',fact_mult_Oconvergent) ).

thf(2351,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)],[365]) ).

thf(914,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__le__iff) ).

thf(4189,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ A ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(302,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_mult__nonneg__nonpos) ).

thf(2170,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Rings_Oordered__cancel__semiring @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Ozero__class_Ozero @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ A ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( 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)],[302]) ).

thf(506,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_abs__leI) ).

thf(2834,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oordered__ab__group__add__abs @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( hAPP @ ( c_Groups_Ouminus__class_Ouminus @ C ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ C @ ( c_Groups_Oabs__class_Oabs @ C @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(1163,axiom,
    class_Rings_Olinordered__semidom @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Rings_Olinordered__semidom) ).

thf(4858,plain,
    class_Rings_Olinordered__semidom @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).

thf(72,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_zsgn__def) ).

thf(1468,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 ) )
       => ( ( ( hBOOL @ ( hAPP @ ( 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 ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( 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)],[72]) ).

thf(1014,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/sandbox2/benchmark/theBenchmark.p',fact_square__eq__1__iff) ).

thf(4552,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)],[1014]) ).

thf(456,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ ( c_Groups_Oone__class_Oone @ D ) ) )
           => ( hBOOL @ ( hAPP @ ( 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/sandbox2/benchmark/theBenchmark.p',fact_power__decreasing) ).

thf(2655,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Rings_Olinordered__semidom @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ D @ A ) @ ( c_Groups_Oone__class_Oone @ D ) ) )
           => ( hBOOL @ ( hAPP @ ( 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)],[456]) ).

thf(852,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/sandbox2/benchmark/theBenchmark.p',fact_mult__left_Ominus) ).

thf(4002,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)],[852]) ).

thf(23124,plain,
    $false,
    inference(e,[status(thm)],[2630,3962,1411,2612,4835,2778,1559,4022,3434,2412,3566,4877,2199,3021,1501,4201,2452,4005,4571,3053,2336,2131,4909,2109,4824,3399,2263,4183,4290,3979,1840,4852,2787,2595,2499,2559,4015,1950,4992,1554,4904,4166,2250,3990,2210,4226,2622,3230,3549,5000,3285,3153,3912,1728,4711,1269,4408,4747,2363,1855,4931,4043,3534,4714,2309,2512,1406,2104,4075,1882,2808,1522,2644,1569,2248,4807,4697,3959,2671,2676,3910,3498,2380,1971,1533,1486,4532,1671,3429,2686,1310,3942,3446,3616,4564,2901,4980,4762,3276,1497,2195,1426,3217,3794,4633,3439,4839,3332,3367,4884,4870,2918,1867,2184,4638,1872,3781,4440,4650,2476,2627,2168,4058,1919,3253,1988,1342,3121,5005,1640,1507,3895,3717,4941,3335,4792,3350,2972,4679,4618,1613,1921,4030,2405,2869,2957,1746,3103,3836,2216,3236,2432,3994,3915,2548,4496,2227,3249,4266,3308,1935,1601,3466,4198,2082,2400,4760,4963,1967,4322,2819,1803,1904,1518,1703,5024,1887,4238,1287,1709,3116,2269,3495,4761,1907,1330,1511,4778,1577,3803,2356,3976,2916,4524,3195,3029,4743,1792,3644,1298,1760,4956,3542,1313,4349,2562,2488,4568,4984,4766,1462,3517,2403,4664,1479,2024,2989,3180,4771,2829,2857,4862,4928,1347,1572,4830,4317,3734,4952,4460,1494,5020,3557,1875,4793,4507,4302,2388,2533,4253,4967,4008,3258,2178,3322,1946,3524,2794,3670,3702,4815,2017,1489,4696,2237,4988,1415,3887,4583,1714,4173,3934,4903,4443,4072,1450,2090,3393,4857,4258,1373,4454,4600,3295,2782,4820,3163,1418,3453,2880,1682,1858,4147,2925,3973,4134,3357,2254,3569,4871,3300,3093,2555,3290,1283,2073,3771,4810,4162,3851,4718,4055,4842,1896,3712,4996,4825,2965,4892,4211,4475,2367,3537,1928,3189,2281,4504,3505,3685,3036,2530,1526,4307,3676,2137,4040,3361,2002,3243,2799,4038,1433,4701,4643,1943,4707,3870,2154,3906,3492,3406,4082,4483,1863,2011,4853,3011,3303,1691,3531,2118,1338,2737,4910,2496,3679,4700,1651,3899,2922,3450,3503,4216,4893,4589,4629,2574,4888,3720,3169,4480,2646,1915,2093,1311,2232,2660,1768,2382,2773,4999,2000,4915,4712,4920,4375,4878,1676,1666,1824,4821,1560,1343,2805,1519,3867,3742,4344,4091,3752,4521,3233,3371,3884,4785,4499,4449,3134,4493,2890,2414,4836,4160,4661,2528,3664,3207,3682,3778,2015,3889,3166,1602,4608,4776,1819,3144,4979,2327,2345,2653,3282,1423,3948,1365,4192,4725,3090,2565,2826,4155,3314,3297,4757,4947,3137,3065,2872,4385,1836,1787,2344,3763,4104,1634,1392,3667,2112,1291,4080,3931,3033,1740,1910,1541,2115,1772,2377,4974,3381,2709,4636,3119,4012,2638,4219,3001,4808,4123,2245,3105,2300,3784,1983,4957,1762,2515,1534,5006,1375,4486,4789,4429,1425,4845,2509,4233,3704,4930,1719,2537,4840,4883,3060,3087,3018,1567,2460,4518,1361,1964,1292,4772,3714,4513,2349,4736,3573,4049,4653,3141,3311,2877,4983,3846,1424,2455,4545,4938,1414,2240,1842,4744,1700,1810,2172,2551,2085,3723,4867,1889,3555,4814,4804,1324,1776,2519,1297,2598,2243,4244,4934,4120,3628,4152,2140,4223,1539,2836,1456,1668,1974,3819,3390,4829,3937,4717,1852,2813,2766,3692,2482,4902,1979,3074,4069,3005,4416,3364,1627,2370,3215,3147,3476,4535,1991,1814,1779,2208,3856,2748,3999,1663,2662,2338,2423,4995,4550,2634,3903,4352,2203,2702,3563,2043,4889,3587,4966,4765,1712,3592,1846,4906,3639,2440,1515,2221,2189,1616,4921,2936,3660,4919,4131,4708,4470,4613,4594,1483,3985,3262,3396,2121,2235,3707,1510,2391,2904,1959,4116,4354,3079,2570,4256,2941,4797,2602,1301,1595,3839,2734,5016,4577,3792,4359,3274,4269,4017,2894,1374,3343,2260,4786,4998,1833,4341,3768,2791,2789,2657,3472,2393,4801,4032,4465,1795,2716,5022,3636,2048,4283,1615,4077,3255,4926,2674,2568,1535,2991,4975,3404,4758,4769,2161,4537,1312,1635,4413,3123,3436,3487,3619,4887,2193,3023,2029,4186,4096,2293,3051,4796,1271,4837,5011,3726,3651,3695,4099,3330,4171,1865,3340,1997,3700,4060,2457,4854,3641,2358,4433,1356,4716,4790,4929,2214,4828,1492,4958,3197,2802,4300,2896,3384,2079,2361,2679,4704,3251,4369,2642,3815,2493,2987,3038,4305,4620,1806,4914,4699,1349,3270,2257,3732,1933,1722,4805,4586,2839,1571,1376,2197,4107,1969,1307,2632,4911,4985,4208,3192,1838,2593,3971,2252,2278,2928,2974,2824,4694,2605,2610,3055,1557,2625,4833,1630,4818,2780,2267,4028,1821,1731,3800,2446,2757,1556,1460,4684,2842,1952,4990,1688,4720,1594,2212,1643,3687,4954,2999,2069,3490,3539,4844,2051,3423,1445,1528,4085,3159,4827,2931,3953,3241,4918,2022,4703,2535,2101,3512,3178,4067,4890,3908,3876,2693,4965,1626,4741,1304,4868,3293,4735,4879,4177,3849,2005,1413,4426,1948,4907,3031,1583,4136,2096,1758,2107,4886,4260,4813,4580,5025,2318,2784,4529,2037,4052,4319,3048,4859,3921,1340,4688,2333,4812,1551,5003,4986,4168,2582,3571,4658,1547,1961,4864,3842,3833,3129,2831,3737,1770,3376,1516,2863,4937,2977,2060,1743,4366,3082,4673,1566,2143,1605,3260,3754,4950,1368,4641,2175,2821,3209,4961,3058,2407,3786,1531,4324,2850,1484,1499,4437,4561,4905,2920,4145,2276,1830,1901,4164,3448,4314,1716,3940,2224,3305,1432,2165,1637,3114,5015,4020,2429,4768,3828,2322,1538,4241,2545,4800,4982,3412,4781,3359,4737,1285,3095,1543,1849,1284,4748,2579,3388,1585,4695,2728,2939,3071,4516,2948,3697,1536,1981,4624,4611,4976,1485,2320,2711,4991,1749,4046,1427,4093,2860,4838,2892,1272,3156,3520,2848,4819,4392,3288,2435,3478,3584,3067,2885,4997,1654,2071,2875,3168,4548,4278,3457,1316,1649,4698,2479,4851,3401,2088,3589,1459,1817,2960,4944,4873,4419,1913,4656,1844,3874,4738,4780,2325,3928,3625,4646,2997,4102,2342,4463,1646,4913,2098,3825,4468,4681,4338,2934,1276,2591,3789,3709,4666,5008,3369,4126,3220,4715,4791,2026,3630,3501,3003,5010,2764,4010,4834,4705,3662,2526,4527,2416,4451,3085,4745,1491,2796,4964,3598,2330,1607,1661,4446,2315,2019,4197,3324,4175,3879,4214,3844,4591,2576,4478,2230,1893,1309,2739,1444,1442,4869,4372,3552,4927,3379,2287,4917,1774,4472,3997,2719,3892,1546,3100,4843,3132,4357,4597,3469,3658,2218,2374,4880,1348,2302,2517,1725,1280,2385,4706,4670,5004,4811,3882,1395,1625,3987,4787,4841,1986,3337,1673,5014,3690,1480,3203,3062,3015,3328,4631,2704,1789,1288,2913,3633,4912,2040,3045,3982,2636,2751,3956,3897,2201,4777,4755,4981,3560,3347,2186,4281,4157,1693,5019,4024,2572,2714,3426,1891,2340,2668,3673,1610,4923,1827,1268,1532,1378,4876,4229,2539,4501,1930,2353,2810,1352,2600,2898,4382,2485,1384,3605,1582,2265,4118,3817,1500,3965,1923,4410,4206,1925,4713,3740,1880,1885,4691,3812,1399,4271,3918,4088,4221,2372,2640,4627,3924,4891,1565,1416,3432,3200,3575,4734,3901,3013,1331,1606,3474,2754,4335,3212,4723,2206,4784,1737,2887,4767,2438,1490,4742,4908,1917,3316,2351,4189,2170,2834,4858,1468,4552,2655,4002]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12  % Problem  : SWW206+1 : TPTP v8.2.0. Released v5.2.0.
% 0.03/0.12  % Command  : run_Leo-III %s %d THM
% 0.12/0.33  % Computer : n025.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 05:53:25 EDT 2024
% 0.12/0.34  % CPUTime  : 
% 0.96/0.87  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.72/1.13  % [INFO] 	 Parsing done (261ms). 
% 1.72/1.14  % [INFO] 	 Running in sequential loop mode. 
% 2.52/1.34  % [INFO] 	 eprover registered as external prover. 
% 2.52/1.35  % [INFO] 	 Scanning for conjecture ... 
% 3.01/1.49  % [INFO] 	 Found a conjecture (or negated_conjecture) and 1271 axioms. Running axiom selection ... 
% 4.10/1.72  % [INFO] 	 Axiom selection finished. Selected 1265 axioms (removed 6 axioms). 
% 5.21/2.09  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 5.33/2.12  % [INFO] 	 Type checking passed. 
% 5.33/2.13  % [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 ... 
% 57.34/14.27  % External prover 'e' found a proof!
% 57.34/14.27  % [INFO] 	 Killing All external provers ... 
% 57.34/14.27  % Time passed: 13739ms (effective reasoning time: 13128ms)
% 57.34/14.27  % 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)>
% 57.34/14.28  % Axioms used in derivation (1265): 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_cnj_Obounded, 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_nat_Osimps_I3_J, fact_complex__cnj__i, 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_Re_Oadd, fact_conv1, 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_Bseq__iff3, fact_diff__add__inverse, fact_abs__minus__commute, fact_abs__if, fact_norm__power, 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, arity_Nat__Onat__Rings_Ono__zero__divisors, fact_nat__less__cases, 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_mult__eq__0__iff, fact_Re_Ononneg__bounded, fact_less__imp__le__nat, fact_le__imp__0__less, fact_add__leD1, fact_ln__eq__zero__iff, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J, fact_add__less__mono, fact_lemma__interval, fact_Bseq__minus__iff, 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_cnj_Ominus, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, 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, arity_Polynomial__Opoly__Groups_Ominus, fact_abs__diff__triangle__ineq, fact_leI, fact_field__power__not__zero, fact_natceiling__le__eq__one, arity_RealDef__Oreal__Rings_Oidom, fact_mult__le__mono2, 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_linorder__neqE, fact_o__eq__elim, arity_RealDef__Oreal__Groups_Omonoid__add, fact_neg__0__less__iff__less, 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_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_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J, fact_abs__triangle__ineq2, fact_Re_Opos__bounded, fact_zadd__left__mono, fact_sum__squares__le__zero__iff, arity_Int__Oint__Rings_Olinordered__semiring__1__strict, fact_mult__right_Ozero, fact_mult__le__cancel__left__neg, fact_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_add__mono, 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, fact_add__right__mono, arity_Polynomial__Opoly__Orderings_Oord, 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_gr0I, fact_sgn__minus, fact_le__fun__def, fact_minus__add, arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add, fact_norm__triangle__ineq3, fact_power__minus, fact_abs__add__one__gt__zero, fact_lemma__NBseq__def, 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_complex__In__mult__cnj__zero, arity_Nat__Onat__Rings_Oordered__comm__semiring, fact_nat__mult__1__right, arity_Complex__Ocomplex__Rings_Osemiring__0, fact_zero__le__power__abs, fact_ord__less__eq__trans, fact_zero__reorient, arity_Int__Oint__Groups_Oordered__ab__group__add, fact_mult__right_Odiff, fact_le__iff__add, fact_one__reorient, fact_diff__commute, fact_minus__less__iff, 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fact_zle__diff1__eq, fact_order__less__imp__not__eq2, fact_Suc__lessD, fact_ln__le__cancel__iff, fact_cnj_Ozero, fact_sgn__less, fact_g_I2_J, fact_add__left__imp__eq, arity_Int__Oint__Rings_Osemiring__0, fact_trans__less__add2, fact_ln__ge__zero, fact_real__zero__not__eq__one, fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J, fact_zpower__zpower, arity_Complex__Ocomplex__Groups_Oab__semigroup__add, fact_add__leD2, fact_mult__mono_H, fact_real__add__mult__distrib, fact_real__of__nat__le__zero__cancel__iff, arity_Nat__Onat__Rings_Ocomm__semiring__1, fact_abs__ge__minus__self, fact_realpow__minus__mult, fact_less__natfloor, fact_order__refl, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, fact_complex__mult, fact_add__diff__assoc, arity_Nat__Onat__Power_Opower, fact_ab__semigroup__mult__class_Omult__ac_I1_J, fact_not__less0, fact_complex__Im__cnj, arity_Int__Oint__Power_Opower, fact_diff__0, fact_ln__gt__zero__imp__gt__one, arity_Int__Oint__Groups_Ouminus, arity_Nat__Onat__Groups_Ocancel__comm__monoid__add, fact_nat__diff__split, fact_norm__one, fact_zadd__0, fact_xt1_I5_J, fact_abs__of__nonneg, arity_RealDef__Oreal__Rings_Omult__zero, fact_real__mult__right__cancel, fact_mult__left_OisUCont, fact_not__less__eq, fact_nat_Osimps_I2_J, fact_ln__mult, arity_Int__Oint__Groups_Oab__semigroup__mult, fact_mult__left__less__imp__less, fact_diff__mult__distrib2, fact_natceiling__one, arity_Polynomial__Opoly__Groups_Oordered__comm__monoid__add, fact_neg__less__iff__less, arity_Complex__Ocomplex__Rings_Osemiring, fact_less__zeroE, fact_termination__basic__simps_I2_J, fact_minus__zero, arity_Complex__Ocomplex__Groups_Ogroup__add, fact_not__one__less__zero, fact_reduce__poly__simple, fact_abs__diff__less__iff, fact_zero__less__norm__iff, fact_CauchyD, fact_one__le__mult__iff, fact_natceiling__add__one, arity_RealDef__Oreal__Rings_Olinordered__semiring, arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra, 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fact_nat__mult__le__cancel1, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J, fact_Suc__mult__less__cancel1, fact_order__neq__le__trans, fact_complex__mod__triangle__ineq2, arity_Int__Oint__Rings_Olinordered__semidom, fact_equation__minus__iff, fact_le__funE, arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add, arity_RealDef__Oreal__Orderings_Olinorder, fact_mult__neg__pos, arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult, fact_diff__int__def__symmetric, arity_Nat__Onat__Groups_Ocomm__monoid__add, arity_RealDef__Oreal__SEQ_Ocomplete__space, fact_le__add__diff__inverse2, arity_RealDef__Oreal__Orderings_Oorder, arity_Polynomial__Opoly__Groups_Oab__semigroup__add, fact_less__antisym, fact_not__real__square__gt__zero, fact_not__one__le__zero, arity_Polynomial__Opoly__Rings_Olinordered__idom, arity_Int__Oint__Groups_Ogroup__add, fact_add__diff__cancel, arity_fun__Lattices_Oboolean__algebra, fact_mult__cancel2, arity_Polynomial__Opoly__Groups_Ouminus, 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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, arity_Complex__Ocomplex__Rings_Oring__1, arity_Complex__Ocomplex__SEQ_Ocomplete__space, fact_diff__diff__right, fact_power__le__imp__le__exp, fact_power__one__right, fact_convergent__const, fact_add__nonneg__pos, fact_mult__less__cancel__right__disj, fact_mult__right_OisUCont, arity_RealDef__Oreal__Groups_Oab__semigroup__add, fact_add__less__cancel__right, fact_Suc__mult__cancel1, fact_mult__strict__right__mono, arity_Nat__Onat__Groups_Ocomm__monoid__mult, fact_real__of__nat__ge__zero, fact_add__nonpos__nonpos, arity_Int__Oint__Groups_Omonoid__mult, arity_RealDef__Oreal__Rings_Olinordered__semiring__strict, fact_Suc__le__lessD, fact_mult_Ozero__right, fact_Re_Odiff, arity_RealDef__Oreal__Rings_Oordered__ring, fact_add__increasing, fact_not__add__less1, fact_xt1_I12_J, fact_norm__zero, fact_abs__power__minus, 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arity_Complex__Ocomplex__Groups_Ominus, fact_mult__1, 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
% 57.34/14.28  % No. of inferences in proof: 2534
% 57.34/14.28  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 13739 ms resp. 13128 ms w/o parsing
% 58.39/14.67  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 58.39/14.68  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------